I consider myself to have AP. Why?
When I did interval tests, I would hear the first note (C), the second note (A flat), then would identify the interval based on knowing the names of those two notes. I don't know what an interval is unless I know the two notes.
When I did chord identification, with inversions, I would identify the four notes of the chord, then listen for the bass note. I would hear B-D-F-G, and would identify it as the first inversion of V7 for C major.
When I did melody playback, I never understood why the examiners played the tonic chord before playing the melody. The notes came time me like words on a piece of paper.
I don't think about intervals and note relationships, all I do is hear the notes and I know their names.
I also have a common issue with other people with AP - I mix up octaves and registers all the time.
By the way, my ricebowl resonates at an E, and my larger soupbowl resonates at a C.
This describes my AP experience almost perfectly. When I first started taking music theory classes, it was at first hard for me to identify all the different seventh chords because I was accustomed to just thinking of a chord as "C major plus a B-flat" and rather than recognizing inversions by their aggregate sound (so to speak) I would have to listen and figure out which note was in the bass and then figure out the inversion from there. I too confuse octaves now and then (usually I'm only off by one octave if any) and I also tend to sometimes confuse perfect 4ths and 5ths because they are comprised of the same notes (6ths/3rds and 2nds/7ths sound different enough from each other that they aren't a problem)
The syllables I hear for the notes are fixed do solfege, but without accounting for different syllables as sharps or flats (G natural, G flat, and G sharp are all "sol" to me but I know by the harmonic context whether there is an accidental - I think this partly explains why I have difficulty with double flats and sharps when playing music in complex keys.)
In general, automatically knowing the solfege syllables was a great help to me in sight singing classes until I transferred to a school that used moveable do. I remember one day we were practicing some sight-singing, but it was written in soprano clef, so I had to transpose the notes in my head to treble clef. The music was in a key other than C, so then I had to mentally transpose it again to make up for the fact that I hear music in fixed do, and then to top it all off, the excerpt we were singing was too high for the students' voices so the teacher transposed it down a few steps! I'm still not sure how I got through class that day - my brain was tied in knots!
I sometimes have difficulty singing a note on command (for some reason the note that comes out of my mouth isn't always the same as the note I'm hearing in my head....) and I sometimes have difficulty identifying the pitch of a non-musical tone (for example if someone asks me "what pitch is that dog barking at?" or something weird like that.)
If a piano (or some other instrument) is out of tune, I will still identify the notes as what they are supposed to be up to a certain point. A slightly flat A still sounds like A to me (just like we might identify light green or dark green colors all as "green"). Playing on a VERY out of tune piano is distracting to me, especially when the entire piano isn't equally out of tune, for example if the bass keys are much flatter than mid-range keys.
I think that possibly having extremely good relative pitch might actually be more useful than having absolute pitch, but at the same time I don't think I would want to give up my absolute pitch! My husband and I were reading along with Messiaen's Transfiguration the other day and it was beyond me how the choir could find their starting notes in various places without having perfect pitch.
