'Hermaphrodite' comes from Greek mythology; he's the son of Hermes and Aphrodite, who got fused with a nymph (because she requested some other god to be with him forever, and Greek god being Greek gods...). The term is also used in the scientific community (specifically, biology) to describe individuals or species who have both male and female genitalia over the course of their lives, though I'm not sure if that definition applied to Hermaphrodite himself (and it definitely doesn't apply to any known humans). (It's also an outdated and offensive medical term for intersex.)
IIRC, 'futanari' means 'dual form' and in the English-speaking world seems to be much more common than 'hermaphrodite' in porn communities but doesn't pop up much outside them. And yes, people have used it to refer to non-hermaphrodites e.g. on rule34.xxx, there are currently over 13,000 images tagged 'futa_sans_pussy', which is nearly 5% of the over 273,00 tagged 'futanari' (which includes all variants e.g. without balls, without pussy, full-package), and I wouldn't be surprised if the actual number was higher due to untagged examples.
So even though the original intent was for futanari to be a synonym/translation for hermaphrodite, an argument can be made (and is currently being made, by me) that it has since evolved into something different. That kind of thing happens with language all the time.
Depends on which definition of futa you're using. If it's 'hermaphrodite' (like you're arguing), then no; if it's 'chick with a dick' (like I'm arguing), then yes. Calling her half-sexed is unambiguously correct, but that's a setting-specific term, so it's disingenuous to say that we can't call her a [insert generic term for 'chick with a dick' here].