I've been through some "discussions" with Carol. First, I admire her contribution to both education and performance. And I think some people are missing the point. Lots of album credits and references did say "Fender bass" even if it wasn't an instrument made by Fender. And it's true that lots of union rolls listed it as "Fender bass" and not "electric bass"- I know that's what Don wrote down when I joined the AFM (Local #102) in 1976. And Carol did write one of the best books at the time on playing this isntrument. The critical factor in her books was that it was an electric bass and she used some stuff from her guitar playing and some other techniques that pertained to the electric bass. Her approach was NOT to just present it like an amplified version of the double bass.
But, like most CK things, there's a certain amount of hyperbole in there too. Of course Gibson, Rickenbacker, Fender, and others called it "electric bass" and "bass guitar"*
BTW, I despise "bass guitar". I don't play it at all like a guitar either physically nor conceptually. When I was younger, more stupid, and less diplomatic, I got tired of asking the leader in one band to just call it "bass" or "electric bass" so one night I left my StingRay at home and borrowed a friends Fender Bass VI. I played "bass guitar" all night long. Chuck Berry double-stop boogies, chords, used the whammy bar, everything like a guitar player. They never called my ax a "bass guitar again".