We started calling them 'presents' patrols since we almost always were passing out soccer balls or gum or something.

Seriously, the reason why we didn't like the term 'presence' was not that it was not in the field manuals; it was because it often denoted a patrol that was routinized, purposeless, and non-productive, if not actually counter-productive.
Spot on. The term presence becomes synonomous with routine interpreted to mean "no planning and no purpose". We did that as UN observers too often and I suspect it migrated from the Balkan operations to US operations.

It is not the same as a presence mission; establish a presence is a mission. Conduct a "presence" patrol however soon turns into drive through/walk through without a purpose, without a plan, and all too often without a clue. We saw this quite a bit at the JRTC especially in the earlier days of MREs; such patrols were meat on the hook for Geranimo

Best
Tom