When I was about 11 years old, my music teacher tried to test me and started playing notes on the piano, expecting me to name them. I knew he was trying to see if I had perfect pitch, but I got them all wrong. Funnily, though, every time he played a note it really, REALLY reminded me of a song; I just didn't know which one. I was a little frustrated about it and decided to practise so the next time he tested me I would get it 100% right.
Over the course of about two months, I allocated an individual song (or phrase of a song) to each note to help me guess it, and it worked! After a month or two, I nailed it and always got it right. At first, I could only do it with the piano. After 4 years I can name each note as soon as it's played or sung and the notes on the piano even remind me of people's voices for some reason.
The strange thing is, I can only do this with isolated tones: If I listen to a song that's not live I can't name a single note! If I'm lucky, a certain chord of a song may remind me of another song, so I can name the KEY of a song, but I can never name notes unless they are isolated. I don't know whether this is because my relative pitch takes over because it's stronger (I can easily tell intervals by relating to other songs I know) or because everything happens too fast. Also, if a song has been transposed even slightly I notice it straight away and it bugs me, but I find that I get used to this after a while.
I know that this isn't perfect pitch because it is something that I developed and learned through "training" but what else could it be? What on earth do I have, apart from relative pitch?