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.

If a card with persist is removed from the graveyard after it dies but before the triggered ability resolves, it won't be returned to the battlefield.

Once Obstinate Gargoyle has been blocked, causing it to gain flying by modifying it won't cause it to stop being blocked.

If a creature with persist that has +1/+1 counters on it receives enough -1/-1 counters to cause it to be destroyed by lethal damage or put into its owner's graveyard for having 0 or less toughness, persist won't trigger and the card won't return to the battlefield. That's because persist checks the creature as it last existed on the battlefield, and it still had -1/-1 counters on it at that point.

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

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

If a creature has +1/+1 counters and -1/-1 counters on it, state-based actions remove the same number of each so that it has only one kind of those counters on it. A creature's persist ability can bring it back again if its -1/-1 counters are removed this way.