This question largely boils down to a question of definitions -- but these are politically contentious definitions. As such, I want to divide my answer into three parts: (a) difficulties with your definition of philosophy, (b) problematic or questionable interpretative choices regarding the "existentialists", and (c) is existentialism philosophy when these things are clearer?
First, you helpfully give us your definition of philosophy:
I believe that philosophy is defined by its method, which is based on logical analysis.
This is an interesting claim, but this does not explain what "logical analysis" entails. Presumably, you're harping to the logical positivists. An immediate question presents itself, why should we privilege this mode of analysis? And then why, even if we privilege it, should we call that philosophy?
It is not merely any discipline that investigates the world's conditions and the meaning of life.
I don't exactly know what you mean by "world's conditions", but philosophy isn't most assuredly is the discipline that tries to evaluate things. If it isn't so, then your first sentence's definition collapses through an identity (biconditional if you prefer) a pretty simple reductio:
- Philosophy is not about finding the meaning of things.
- Philosophy should do logical analysis.
- Logical analysis looks at the meaning of things (i.e., the value of things is that they should fit under logical analysis).
- Ergo, philosophy is not about logical analysis.
To be a philosophical idea, that idea has to be based on reason
Sure, but what does reason mean here? Is it a pure synonym for "logical analysis"? If so, then what justifies that choice? Can logical analysis itself justify itself? (your dictionary definition suffers from the same ambiguity.
First, regarding the thinkers you categorize as existentialists, the term "existentialist" is mostly a term coined by Sartre and applied retroactively to Kierkegaard.
Second, you state:
the existentialist philosophers - Sartre, Camus, and Kierkegaard especially - had intuitive rather than rational justifications for their claims about life and ethics.
This seems pretty dubious to me. Intuitive means depending on intuitions, but that would be closer to say Reed's common sense philosophy or the empiricists than to Kierkegaard or Sartre (I don't know too much about Camus).
They were not interested in building a logically constructed philosophical system, but in expressing ideas about life as it is lived.
This again is a bold assertion and interpretation. Depending on what is meant by "logical constructed system" (avoiding the circularity in your claims of calling that "philosophical"), you might be right that they didn't see the point of doing philosophy as trying to break the world down into analyzable bits and pieces.
But the second half of the claim ( "but in expressing ideas about life as it is lived" ) seems to be much more dubious to me as a description of their philosophies. I'd be interested to see it defended from the text. Sartre and Kierkegaard do start from the subject -- but that's not the same thing as trying to engage in a description of "life as it is lived."
Instead, I take it that both accept Kant and Hegel's idea that we cannot escape our cognitive (and rational) apparatus well enough to do pure analysis. Kierkegaard, for instance, does not deny that there are objective truths. Instead, his point is (greatly simplified) that we don't have limitless access to these and we are limited beings.
In terms of your comment about Nietzsche, Nietzsche speaks in many poetic ways, but his critique there is actually quite close to the logical positivists. The point being, that it is slavish (i.e. without reason) to derive morality from the instruction of others rather than from your own reason and power.
As I stated above, I do want to address your conclusion in the question and the thought that existentialism should not be called philosophy.
First off, if we grant your definition of philosophy, i.e. "logical analysis" and fill out what that means, you're probably right -- existentialism should not then be called philosophy. But if what you mean parses to "the application of symbolic logic to claims" and the consideration of things insofar as they are bound and unbound variables, then all sorts of other things we regularly consider philosophy aren't either.
On that definition, the first philosophers appeared in the late 19th century, because the methodologies prior to that were not largely based on symbolic logic. That seems problematic.
Given this dichotomy, it seems best to reject this as the definition of philosophy. Why, then, would someone assert this definition? Clearly, the goal is to exclude certain methods and views from the realm of philosophy.
A better definition might be either the tradition of thinking that grows out of Plato. This would put most everything we've thought of as philosophy over time into philosophy. Maybe in the 21st Century, we can give it a less Euro-centric version and refer to the equivalent considerations in other traditions as well.
If you want to heap opprobrium onto "existentialists," writing them out of philosophy by definition does not seem very viable to me. I would instead suggest the following -- instead, assert that you think "existentialists" do philosophy poorly, that they get caught up on immaterial things and don't focus on questions that grasp the nature of reality well. And then argue as to how this is so (preferably in a non-question-begging manner if we're trying to do philosophy).