I think if not fleshed out in the mythology being used in the setting, it’s in the DM’s prerogative. If I were DM, I’d say the Phoenix has to actually die before it can respawn.
Slicing off a Phoenix’s wings will just result in an injured and very pissed-off Phoenix.
Moreover, I’d stipulate that whatever Phoenix parts (butchered, raw, or cooked, or even partly-digested) would disappear whenever that Phoenix respawns. And for a spicy twist: someone who digests any part of a Phoenix will have a psychic link to the Phoenix. Wisdom check after every long rest (three days after ingesting the Phoenix) to determine whether or not the person retains control of their body. Failing this wisdom check thrice in a row results in the Phoenix gaining complete control. Succeeding this wisdom check thrice in a row results in the person regaining complete control of their own body.
EDIT:
Thinking about this more, I think this can be fleshed out even more. There is only one Phoenix, which was eaten by a bunch of people believing eating it would result in gaining whatever powers the Phoenix originally had, maybe being impervious to fire. However, the Phoenix took over their bodies instead. Many many many years later, the Phoenix never really dies: it just choose a body it controls, and transforms it to “its original body”. Thus, now, the Phoenix is known for its “immortality”.
I use Anki to keep my French and Japanese from deteriorating any further. Why those langauges? For Japanese, it’s the usual answer: anime and manga. For French, I had to learn it in school and I just don’t want to let it be forgotten.
A good number of the decks I use for language learning, I made for myself. Those that I’ve gotten from AnkiWeb were already gone, but here’s some similar ones:
French Sentences — a collection of French sentences, sorted from easiest (most simple) to hardest (most complex). This deck begins with very simple sentences, and slowly gets harder as you progress.
I’m pretty sure this is an exact, if not a close copy of the one I got a while back.
Dictionary of Japanese Grammar - DoJG — a deck made from the entries of “Dictionary of Japanese Grammar” (basic to advanced).
My copy of this deck has undergone significant changes in card design (and function, like showing a random selection of the example sentences), but this one should be similar.