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".