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.

Every patrol should have a purpose, or multiple purposes. It can be 'engage citizen X and Y', or 'observe activity in marketplace', or 'conduct route reconnaissance'. Too many times, 'presence patrols' were labeled that way because subordinate commanders had no clue what they should be doing or were too lazy or too busy to properly plan their daily activities. A 'presence patrol' was easy to mount and easy to count as successful. In our corner of the world we just directed that they be called 'patrols' and that each had a clear task and purpose attached.