I don't have perfect pitch (never trained in it) but I've played violin a lot (started in college) and I can hear the G-ness, D-ness, A-ness, or E-ness of a note. I do have to be actually trying to listen though, except for if they are G3, D4, A4, or E5.

So I have the latter, but not the former (i.e., no perfect pitch). The difference is, I can choose not to observe it. It reminds me of how I've studied Japanese for 15 years, but I can still sometimes choose not to read certain kanji if I glance at a legible word — the same is not true for English, if I see something even for a split second I've already read it.