I believe my claim of having perfect pitch stands on solid ground, and consists largely of what OP is describing. That's because I can demonstrate it. I wouldn't go so far as to say that the described method is a reasonable way to _acquire_ perfect pitch, but I do contend that what is described is a reasonable method of demonstrating and explaining one's perfect pitch. If you do have perfect pitch, that is reflected in the fact that you don't transpose the music you hear on in your head.

This idea of recalling songs is an easy way to explain perfect pitch to others in a way they can easily try for themselves. If you can sing the first note to a song, and then afterwards you actually listen to that song, and you are actually in tune, then you have perfect pitch. That's a far cry from being able to name the note, or rattle off the notes in a chord, but in my view the mental recall is the true innate essence that undergirds every other component of the umbrella of perfect pitch.