Is the ability to "remember" and reproduce a pitch independent of perfect pitch?
I feel like I have the former but not the latter. I can't say what a pitch is immediately on hearing it, but I can back into it with some thought.
There are some different skills: perfect pitch (context-free accuracy), relative pitch (an interval, given a starting point), memory (what you described), enough accuracy to keep from drifting, blending (when in a choir context), etc., that are unique to vocals.
It is, and it's not something that everyone has. I do, and it's a wonderful skill. My mom was amazed when I demonstrated that I wouldn't need a backing track when I volunteered to fill in for a singer at my grandmother's funeral. I remember that song and its pitch, and that's all that I need.
My mum’s contention (she is very musical) is that perfect pitch is learnt. One has memorised the pitch. But that it’s also a hard thing and rare to be able to memorise.