Maybe it’s a kli shemelachto leheter–after all, you use it to look at the time 50x a day, and use it for apps only a few times a day.
And even if it a kli shemelachto le’issur, maybe you’re always wearing it lezorech gufo umekomo, since you always need to know the time.
About the motion detectors–who says you’re allowed to keep them on on shabbos? Perhaps there are heterim to go to places where they happen to be on, but to davka keep them on at home on shabbos?