I always thought that it was a harmonic when you touch the string lightly (say at the 12th fret) and it was a false harmonic when you turn you pick and produce one like that. (that thing that Zak Wilde can't seem to stop doing)
Now if those do have different names, then what do you call it when you fret a note, then tap the string 12 frets up (for example) to get that kind of harmonic?