MAITLAND: Doctor, is there no way you can get into your ship?
DOCTOR: No, not unless they return what they stole from my ship.
RICHMOND: But will they'll never give it back to you!
DOCTOR: Oh my dear, they're not invincible, no-no-no. Did any of you notice the peculiarity in their eyes?
MAITLAND: No I can't say I did.
DOCTOR: It's a fallacy of course that cats can see in the dark, they can't; but they can see better than we humans because the iris of their eyes dilates at night. Yes...huh! Haha!
My thought has always been that the first doctor was lying so long while pretending to be human that he just didn't completely weed the habit of referring to himself as human.
As far as the TV movie, my head canon is that he was expressing either a kinship via association, or referring to similarities between his species and humans. Either of which could be a good in-universe reasoning too.
It isn't like the doctor has to speak truth at all times. He's often shown misleading people, flashing psychic paper to outright fake identity, etc. He could absolutely make claims about being human or part human and it wouldn't be inconsistent for him to be lying for some reason.
Rule #1: The Doctor Lies.
But that business with the Doctor being half human needs a bit more explanation than what you have: there are several instances of Paul McGann's Doctor using the fact that he's half human as a plot device. You'll need to make your peace with that in a more creative way.