An Aura controlled by another player does not cause a creature you control to be modified.

A creature with a counter on it is considered modified no matter what kind of counter it is or which player put it on that creature.

A creature that is equipped is considered modified no matter who controls the Equipment that's attached to it.

You can always activate an ability that will cause a creature to adapt. As that ability resolves, if the creature has a +1/+1 counter on it for any reason, you simply won't put any +1/+1 counters on it.

If a creature somehow loses all of its +1/+1 counters, it can adapt again and get more +1/+1 counters.