Perfect pitch is being able to name any individual note and to hear when they're not quite in tune. Although it's also a curse, because perceived pitch changes with age, so you have to keep learning to compensate. And domestic acoustic pianos are often tuned a semitone flat, which really messes with your head.
Advanced pitch recognition is being able to pick out the notes in a complex chord in open form.
Professional pitch recognition - as demonstrated by some conductors - is being able to list the pitches in any random set of notes played at the same time on a piano. Including notes in the lowest octave.
It's a lot easier to get to the last two if you have the first.
I once heard a C on a piano and played it a few times and then remembered it. I thought it was absolute pitch.