Your question is very interesting since there is no conclusive answer to your query. I have personally experienced what you are asking about during my student's years at the university and came to the conclusion that at the current level of development of biology your question can't be answered. Actually, that it is rather a matter of personal choice, than of exact science or philosophy. I myself had the stamina to ask dozens of scientists in my home country what do they think about the definition of Life and I almost got as many answers as people I had asked. I studied molecular biology and from experience with precisely the effects of the question you put can tell you most biologist run away from the issue unless it jeopardizes in some way their own research. Then it is just the opposite-they find this point of view in philosophy which best suits their own personal agenda and stick to it. For example, virologists tend to think viruses have 2 forms-inactive which isn't alive outside the cell and active, which is alive inside the cell. A biochemist would tell you anything having metabolism is alive, so we should always look for the metabolism first, while a geneticist will tell you anything alive has DNA coding for proteins. I want to stress this is my personal experience, so I am not talking about what any scientist should say from a philosophical point of view. I am telling you what they say in reality.
Philosophers, however, are an entirely different stock :), if I can put it that way. They have their own predetermined answers based on their own schools of thought. If you look at the previous answer you will find what the analytical school of thought has to say on the issue. They tend to have one very "strict" picture of reality as science "leading the way" towards knowledge and philosophy just "polishing the terminology" after science so everyone can agree on what exactly they are looking at and what exactly is this or that. Then, yes, the answer is the definition of Life is a question in the philosophy of biology, which is a branch of the philosophy of science, which is a branch of metaphysics. And all the other questions pertaining to it are part of the same classification, too. This is your "standard" dictionary answer. If any biologist sticks to it that means he or she isn't interested in what exactly Life s and is giving away the right for an answer to philosophers. Then the definition of Life and all other questions(questions that can't be answered by experimental evidence)are matter of particular "philosophies of..." that particular field and you get one multilayered structure where the philosophy of this is part of the philosophy of that, which is part of the philosophy of.... and so on until you reach philosophy of biology. This is what I(again personal opinion based on personal experience-anyone feel free to disagree)see when I look at western sources about theoretical biological questions. There is a strict structure in this philosophy where everything pertaining to philosophy is divided in 3 big branches-metaphysics, epistemology and ethics and every question is carefully analyzed(this is why they call it analytical)and put in either branch. Then they try to find a subbranch where to put it if there is that need and if further need arises to put it in other branch and so on until it finds its proper place. This is why the question what is Life is a question in the philosophy of biology, are viruses alive-in the philosophy of virology(or more generally-microbiology), is Von Neumann machine alive-in the philosophy of Artificial life and so on. It is the strictest philosophy I have seen and here everything has its predetermined place and domain and questions are analyzed so they can fit in with experimental data.
However, this is not what everyone agrees on to be the structure of philosophy in the world.(I feel like sometimes Americans just forget they aren't the center of the world :)Besides the analytical tradition there is also the continental tradition as well as a number of other philosophies dating back to ancient times which view the world in a different light. There are myriads of ways philosophy can be divided and each of them can produce different branch where the definition of Life can fall under. But for the sake of the discussion I will concentrate only on what I have experience with, not what is theoretically possible. As I told you if you keep up with the analytical tradition you will end up thinking the definition of Life is under philosophy of biology and it respectively, under metaphysics. But if you go into a philosophical department in my country you won't find a faculty of metaphysics, epistemology or ethics. You will find faculties of ancient, medieval, enlightment, modern and so on philosophy(and may be one of the philosophy of science as an exception of the rule). This is because the study of philosophy here is divided by the age and more rarely-by the subject the particular philosopher is working on, rather than some "great" classification of philosophy like it is in the USA. Actually, there are some philosophers here who would argue such a division is impossible or at least harmful for philosophy in general because it is a discipline of many interconnected opinions, that is why when you start tackling one question in particular you will always end up reaching others too. They would argue everything in philosophy is so heavily interconnected it is impossible to put a rigid classification on philosophy and to "force" any particular question in one branch or the other. They will laugh at the attempts of the analytics to prescribe any question to a predefined "branch" and would instead insist philosophy is a free endeavour, so the best way to proceed is "to let it go" and see where it will end up. In my country philosophical papers are made by comparisons amongst the philosophy of different philosophers in different times or more rarely-by comparison amongst different schools of thought(and therefore the 3 branch classification is presumed to be only one option used by the analytics). In such an environment the definition of Life question falls into a kind of "historical" analysis as to what different philosophers had to say about what is Life through the ages and comparing who got the best match to modern biological evidence. There under such a methodology there is no branch under which the definition of Life falls into since such an investigation can be made for every philosophical question and philosophy is regarded as some kind of a personal or cultural point of view, rather than a "science-like" discipline where rigid classification can be implemented. There is even that joke around here-that Americans made the 3 branch classification only to be able to divide the desks in their departments so no one could feel offended by the division :) (By the way this is why I don't trust the SEP to give you tangible answers to your queries.) Under such circumstances the definition of Life becomes as much a scientific endeavour, as it is philosophical one and we the biologists are welcomed to join in so to analyze which philosopher got it best right. then the definition of Life becomes an open problem in biology itself and we biologists can propose definitions in accordance with our own research and theories to make together with philosophers a definition which best fits current experimental evidence as to what Life can be and the dominant current theoretical constructs in biology. Then the definition of Life falls under biology as much as philosophy :)
There is still another approach to the problem-to consider philosophy as comprised not of branches but of "schools of thought"(here I am literally translating a term in Bulgarian so I am not exactly sure what the proper English translation should be). Each school is build around a premise holding true in all cases touched upon by the school so a comprehensive view of the world can be build upon it. For example, skeptics would consider the question of a definition of Life as an impossibility since there will be no way everyone to agree on the same definition(in an extreme variant of skepticism)so the quest is utterly useless. Platonists can consider Life to be a form just like number so we can always see Life around us but never be able to isolate the particular property defining it. An extreme optimist would consider all definitions of Life leading to more the less the same result so it will be the definition for him/her. Pragmatist can consider the question to be related to latest discoveries in biology itself so the philosophical definition of Life should follow closely with what biologists are "cooking up" in the labs and so on and on. Since every school of thought can be extended to a complete picture of reality the question what is the definition of Life, to which branch it falls under and what pertains to it is pretty much the question what is the worldview of that particular school and how they imagine what the relationship of biology to philosophy to be. In the end however their approach can be very easy to compare with the one of the analysis of the suggestions based on the views expressed by different philosophers, historical periods or
I myself facing the very same problems you talk about adopted the strategy that the definition of Life and the questions pertaining to it are at the intersection of biology and philosophy and it is better to remain there. Facing the hypocrisy of my colleagues scientists I decided it is best to allow philosophy to "lead the way" so I don't follow under any predetermined direction caused by my personal preferences to one definition or the other like all the other scientist do. I saw that although the definition of Life may not look like a topic of particular interest to them in reality each specialist would defend this point of view closest to his or hers specialty and disregard all the others. As I told you in the beginning each one has an opinion clouded by his or hers job. To be able to escape this trap you need to be able to distance yourself from any particular field so you can see the big picture clearly. At the same time relying on philosophy alone is never enough because as I told you all philosophers can do(at least here under the premises we work on)is to compare different points of view with each other to create something useable in science. But simply following one tradition or the other or pertaining to one school or the other isn't going to do your experiments so in the end I simply decided definition of Life is a real tangible biological(not philosophical)question and should be treated like one! But not by any particular branch of biology which would want to turn it in its favour but by a new general and that implying also fundamental theory of biology capable of reflecting on any question in biology, not only on those found in a particular field. And this is pretty much what I am doing to this day :)
There is this analogy here in my country-philosophy should be like a lighthouse showing the way to what's science, rather than "sterile" field of research rearranging the words in such an order so science can see "the cliffs" in front of it and find the best way to escape them. I fully support this view :) It means philosophy should always be first "paving the way" for science(a point of view often accepted in continental philosophy)and showing it "the cliffs and underwater reefs" so when science comes in it can steer its way through. To this end first comes the theoreticians who use what philosophy can give us to make theories able to explain those phenomena already touched on by philosophy and create new experiments to testify for their validity and the entire experimental instumentarium whose real is to see how much of the joint venture of philosophy and theory is actually true. This is the way I am in. According to this path the definition of Life is as much a branch of philosophy as it is of biology and it is a valid scientific question but you, of course, have the right to choose your own answer to the question :)