I think it's not so cut-and-dry. As a violinist, I can recognize absolute violin pitches, but can't do so with other instruments. This is definitely a learned skill; presumably there's something about the timbre of the different pitches that my brain has memorized after so many years of playing.
However someone with "true" perfect pitch is able to detect pitches across instruments (or even a sine wave) and can distinguish minute differences in pitch; for example, someone with perfect pitch can tell that an instrument has been tuned to A415 (a typical tuning for Baroque music) instead of the standard A440.
The difference between A=415 and A=440 is not "minute" at all; it's a full half-step difference!
Also, no: absolute pitch does not necessarily confer the ability to distinguish very small differences in pitch. Many people with perfect pitch can name a given note if played, but can't reliably identify if the pitch in question is 5 or 10 cents off. This is a common misconception.
Similar thing here. I'm a pianist, and while I don't have perfect pitch, there are certain piano chords that I can pick out and definitively say, "That's an F major, root position, with the root just over middle C". But other chords or even individual notes – I'm not able to distinguish them in that way.
However someone with "true" perfect pitch is able to detect pitches across instruments (or even a sine wave) and can distinguish minute differences in pitch; for example, someone with perfect pitch can tell that an instrument has been tuned to A415 (a typical tuning for Baroque music) instead of the standard A440.