1. The ability to immediately identify a pitch name from listening. This is the common definition. It is analogous to a native language speaker recognizing a spoken word -- it's automatic and requires no effort.
2. The ability to /recall/ a pitch from memory accurately. When you sing a tune, are you singing in the same key as the original recording? Some people can't do this.
3. The ability to identify a pitch name from listening, not immediately/reflexively but by mentally matching it against a well-remembered reference pitch. This requires (2) and also requires some work on your part, to remember a "catalog" of reference pitches.
4. Ability to recognize a chord type and root, either automatically (like #1) or by mental reference to some well-remembered chords (like #3). Of course, the root can be ambiguous and tricky to find.
I can't do #1 (immediate automatic pitch recognition) but once in a rare while it happens. I can do #2 easily -- mentally recall or sing any remembered tune within probably 25 cents. That means I can tune a newly-strung guitar, e.g., pretty well (though not perfectly) by just remembering a reference low E (first note in Airbag) or drop-D (first note in Everlong). I can do #3 with some mental effort -- requires me remembering my catalog of opening note for a handful of songs/pieces, and then doing a slow mental match.