I think it's the former. According to section 6.8 they flip to active at the end of the turn in the provinces you govern. Limes gives you the benefit that the invading barbarians are inactive during the support phase and will not degrade the support in the province they are in for that turn. You'll still need to deal with them next turn.
Correct - Limes basically give you a one turn grace period to deal with a barbarian invasion.