The question of what constitutes "Logical Necessity" might be usefully approached starting by asking which of the two words in that composition takes priority. That is, are you interested in what is necessary, given logic, or are you interested in what is purely logical, given necessity?
In some ways, the idiom "Necessary by Definition" suggests the second, in that you have a modality over meanings. As Conifold notes in their comment, this is often thought of as Analyticity, and I quite like Rudolph Carnap's interpretation of this as operating within the language of scientific practice. Scientific practice, it is held, works in a logically rigorous way, and its language is similarly logically structured, but the modality involved here is about statements/propositions and their connections to observations of the world.
But it seems strange to call these statements themselves the logically necessary ones; you need to have meaning postulates, to have connecting bridge principles which tie propositions to things in the world. As such while analyticity has the flavour of logic, it is still not itself entitled to claim the full ontological status of logicality.
If we have a prior understood modality at work, it may make sense to interpret the logically necessary statements in terms of a logic of those modal operations. It is not logically true that all men are bachelors; however, given an interpretation of how analytic necessity, the universal quantifier and implication connect together, it might be logically true that if all men are bachelors, and if an object is a man, then that same object is a bachelor.
In order for this to work, we need to understand the language in question as having a logical structure characterised by connectives like "for all", "if", "and" etc. And, as Quine has argued, it is not simply a foregone conclusion that the language we have is indeed so structured, and you can't fall into the trap of begging the question against yourself as to the mathematical soundness of your language to be scrutinized. Not even mathematical language can characterize itself without significant difficulty - see e.g. Hilbert, Tarski, Godel.
So perhaps the other avenue might make more sense. Starting from logic, what kinds of modality might we introduce, and what necessary truths might fall out from the logics of necessity we establish?
This is the direction Saul Kripke took Necessity in the field of Modal Logic. In this, necessity is not so much as a prior given over which logic might be run but rather as a mathematically characterisable functor (or, more accurately, a plurality of such functors) over the domain of logical statements. The idea is that Necessary is just an operator, like "if" or "and", that can be built into a logic and using which we can describe some particularly rich structures of true statements. Or, you know, value 1 statements, or statements admitting of multiple valuations etc. etc.
Some such structures will admit of statements that are true purely in virtue of that logic - in fact, some such structures will admit of statements featuring the necessity operator that are true purely in virtue of that logic. These seem to be to be a more accurate place to be leaning on our understanding of Logical Necessity.
However, well, what functor of necessity are we using? Different interpretations of the modal operators may yield different accounts of the purely logical necessities.
Fortunately, in mathematics, we don't necessarily care a whole lot about whether a model is practically applicable in order to say it exists - each of these concepts of Logical Necessity can be modelled, and so each may be worth investigating as a legitimate object in its own right.
But perhaps we still haven't quite got there. It's not so much that we're interested in how to model necessity in logic, but more what the modality of logic itself is supposed to be. What is it in virtue of which we say that from
B - what is the modal force of this kind of "following", this consequence?
A particularly interesting route into this discussion is to look at disputes around the applications of competing logical principles. We create characterisations of different logics that (for example) allow different patterns or principles of inference, and warrant different strengths of conclusions. Are there some statements in the world that are neither true nor false? What can we conclude when we introduce premises that might be both true and false? Do all of our conclusions have to be derivable in computably many steps from base deductive principles?
I am an advocate of Logical Pluralism - I think that "logic" is not an unequivocal term, but rather appeals to different structures of inference and interpretation depending on the domain of discourse. The domain of Mathematics does not admit of fuzzy truth, but the world of Politics and Commerce most definitely does, and Data scientists (my particular field of interest at the moment) need to be aware that one cannot straightforwardly infer conclusions using a one-size-fits-all model of logical consequence.
But the logical tools that we've created in the mathematics can still be useful and informative across domains, even if we deny the possibility of a singular catch-all analytic concept of logical necessity. Try some models, see what they tell you, see if that holds, and refine and keep trying. We're probably not going to get it exactly right, but that's the nature of the scientific method, I believe!