How many rounds does a spell with a duration of 1 minute last?

If a concentration spell lasts for one minute (60 seconds, 10 rounds), when does one start counting down how much time it has left? Does the round you cast it in count towards how long it stays active?

First example, what if I use Aura of Vitality, which uses an action to cast then a bonus action can be taken to heal a target? The spell lasts for 1 minute, or 10 rounds. So on the turn I cast it I can use a bonus action to immediately use it, then do I get 9 more rounds or 10 more?

What about spells like Moonbeam, which don’t take effect on your turn? Moonbeam activates when a creature enters it or starts its turn there. If you’re first to go in a round then you’ll get 10 uses out of it, but if you’re last does that mean you only now get 9 uses out of it since it acts during an opponents turn?