(A bit difficult to describe what I mean without sharing my definitions, but I am cautious of posting them since this is regarding my thesis.) I had to provide a definition for a term I came up with. Since it is a complex matter, I provided one definition and then wrote "alternatively put:" and gave a second definition that describes it a bit differently and more elaborate. My advisor just put a question mark. I thought of merging both definitions together, but they are more or less different "views" of the term, so it didn't make sense. Is there an academically adequate way for me to keep both definitions of the same term?
If they are "views" then they are hardly "definitions". A definition should be precise. If you want to give two statements, which might occur in math, you need to be able to prove equivalence.
If they merely overlap in meaning then there may be cases in which one applies and the other doesn't. In informal language that is probably fine, but in a dissertation that is unlikely to be the case. It is hard to "define" goodness for example.
Note that something like a metaphor or analogy isn't a definition. A definition needs to distinguish what is and what is not conformable.
I'll go along with your advisor and put a question mark.
You can choose one, perhaps, or unify them formally, but two is too many.
If the same term is used in your field in two distinct ways then it may be defined differently for different aspects. But in such a case you will need to be extremely clear when you use one definition or the other. What is it that distinguishes them?
Echoing @Buffy's good answer, and emphasizing a certain point: even if there is no truly "correct" order of logical development of the ideas (meaning what is a definition, versus consequence of definition), you probably need to pick one... Then perhaps prove that another "definition/characterization" is equivalent, or, possibly, is equivalent when both characterizations apply(?).
After proving such an equivalence, then we can forget the (possibly artificial) logical ordering that we used at the beginning (to avoid circular reasoning...)
Here is a familiar example of how to write equivalent definitions,
Lemma 1.3.4: For any set $A \subset X$ the following two conditions are equivalent:
(a) The complement $X-A$ is open in $X$.
(b) Any sequence $(x_n)$ in $X$ with limit $x$ that has all $x_i \in A$ also satisfies $x \in A$.
Proof: Yadda yadda yadda QED.
Definition 1.3.5: The set $A \subset X$ is said to be closed in $X$ to mean it satisfies one and therefore both of conditions (a) and (b) above.
I guess you can give multiple definitions, as long as you then explain why there are multiple definitions. For example, I can imagine if you are talking about "fatigue" you could say "for engineers, fatigue means... But when sports scientists use the term, they mean... The difference is interesting because...."
Basically, you don't want your readers to be puzzled by what's going on. If your supervisor (who is an expert) is puzzled, then everyone else will be too. Have a bash at rewording it, and run it past your supervisor again.
What field is this? If you're doing qualitative work, it wouldn't be out of line to provide an elaborated definition and a more concise (TL;DR) version of the same concept. The shorter definition may be shorthand for the former. The shorter version could also be a more abstract concept that would meet with wide agreement (and be easy to grasp, maybe even by non-academics), while the longer version is a detailed description of exactly what does and doesn't fit into the concept. Indeed, when doing quantitative analysis in the social sciences, it's very common to give a conceptual definition, followed by a definition that operationalizes the concept for the analysis in question.
A definition must be comprehensive and unambiguous.
If you need a second definition, it can mean that your first "definition" is still ambiguous, that you still need to be more precise. In that case, it means that the first "definition" was actually not a definition. So it would not be okay to have two definitions. Work harder on the first definition.
Another case is if you have readers from very distinct fields who have distinct jargons which makes that some of them who aren’t familiar with the jargon of the first definition (assuming it is a true unambiguous definition) don’t really understand it. In that case, a second definition can be interesting as long as it’s just a sound paraphrase. But the best solution remains to find a definition whose words are understandable by everyone.
TL;DR. It’s always a bad idea to have two definitions. However, it can be interesting to present equivalences. That is the fact that one term in a specific field corresponds to another word in another field. This can allow to jump from one field to another field and connect knowledges together.
If you want to help people better understand what your object is by explaining things about it, by stating some consequences of its existence, or by telling its origin, etc. it doesn’t stand in the "definition" part but in the rest of your text parts (etymology, origin, consequences, corollaries, theorems, etc. as it depends on your working field).