In response to "what's wrong with the term "female bodied" and/or "male bodied?" And what's pereferable?
(Not because it's politically correct, but because otherwise it'll sound like you don't have a clue what the fuck you're talking about.)
The usual term, in my experience, is "assigned." i.e. "assigned female;" "assigned male." It's been around since the 1990s, introduced to most via Bornstein's Gender Outlaw. It puts the responsibility for the right or wrong gender label on the birth/medical system's crapshoot prognostication of our future gender based on a glance at our genitals.
"Socialized" doesn't really work as it kind of overlooks that a lot of people, both trans and cis weren't socialized as you might expect. I've gone out with cis femmes who were raised as boys (Mum and Dad thought it was a good idea). I know people of varying genders who were raised on dyke communes. And a lot of trans people were *not* included in the usual run of socialization - they didn't get to play with the boys or the girls, being told instead "get out of here, dyke/faggot/tranny." If a cissexed butch was misread as a boy, people assume she's "socialized female," yet if a transsexed woman was misread as a boy, people assumed she's "socialized male." This makes no sense, and often it's just a thin vehicle for cissexism.
Go ahead and use "socialized as" if you actually know how someone was socialized. Check with them first.
"Bodied" doesn't work as it tends to lump in people across the trans spectrum with the wrong damn body, regardless of how they feel about their body or what's been done to it with or without their consent. If an FtM guy has boobs, he's called "female bodied." But if he has moobs, he's still called "female-bodied." And if he's had chest surgery, he's "female bodied." Yet if your (cisgendered) great-uncle has boobs or moobs, he's called "male bodied." under this system, transsexual women get called "male bodied" while women with AIS (who have an XY set and no uterus... much like a transsexual woman) are called female bodied. And again, intersexed people get rendered invisible. As commonly used, it tends to just further cissexism.
As above, you can refer to someone as whatever-bodied if you've checked in with them as to exactly what kind of body they have, and what they're comfortable calling it.
If you need to talk healthcare, then just be specific. "Anyone with a cervix?" "Anyone with testes?" "Anyone with breast cancer."
"Biological" doesn't work, as it has very strong roots in the assumption that that one (cissexed) person's gender is more natural/biological than another (usually transsexed) person's - this usually builds on the assumption that that which is read on the body by patriarchal medical science ("the body" in this case which somehow does not include the brain) near birth is more legit than that which is slowly carved into it with years of painful work . It also it implies that if you're not a biological whatever, than you must be a... replicant or something.
Being a product of the Tyrell corporation, this doesn't bother me, but I understand some of you fleshy humans get titchy about it.