Then I did some searching and found multiple examples of both definitions in use, making things murky.
So I turned to Merriam-Webster’s dictionary: “ of, relating to, or being a vulnerability (as in a computer or computer system) that is discovered and exploited (as by cybercriminals) before it is known to or addressed by the maker or vendor”
And of course they use an “or” to make it ambiguous as to whether the days start counting when the vulnerability becomes known, or when the vendor has addressed it.
No, the full name was always "zero-day exploit". The number 0 refers to the days between the vulnerability being known by the vendor and the public availability of the exploit. So the vendor has zero days to create a security patch before the release of the exploit.
The term "zero-day vulnerability" is a derived term to refer to a vulnerability affected by a zero-day exploit. Similarly, a "zero-day attack" is a derived term to refer to an attack carried out using a zero-day exploit.