Few things have a "set meaning" because that's how language works. Meanings are consensual, not static nor existing in some abstract, pure form. Any identity one claims is not necessarily a precise description of who they are--I know several people who, in fact, do not find any of the existing gender or sexual orientation designations precise enough to fully agree with, but they'll claim the one that's closest because it's preferable to have some identity than be left in the uncomfortable position of not having a term to identity with at all.
The issue with "male" and "female" is not a lack of precision but rather their narrow scope and insufficiency for describing various realities. There is a difference between having to use a term you don't identify with at all and using one that you don't completely identify with but find close enough to suit your purposes.
This is why "male" and "female" (or, if you prefer, "man" and "woman") get modifiers. Designated male/female at birth, trans woman/man, etc. In and of themselves, they don't necessarily say enough to constitute something one is comfortable identifying with.
Basically, the purpose is for people to have terms describing identities they are comfortable claiming as their own, rather than forcing them to fit inside boxes others have built and insist on keeping narrow and limited. I may be explaining it poorly but that's where I'm coming from. (I say this as a hetero cisgender male, so I have no personal stake other than my friends who do.)