I'm pretty sure (though not 100%) that moving into an area containing neutral units counts as your "one battle" even if the result is retreat or alliance. Only one POTENTIAL battle between both moves that Mobility gives you.
This is incorrect. It only counts as your one battle if you either don't use diplomacy at all, or you end up with a "red" result and opt to battle. If you usea "gold" or "grey" result, it does NOT count as a battle, nor does it count as a battle if you use a "red" result and choose to exercise the retreat option.
So you could use your first move to attack an enemy, and then a second move to attempt diplomacy against neutrals. Note, however, if you do that, and your diplomacy result is "red", you MUST retreat, since you would not be allowed to start a second battle.
You could also do diplomacy first, and as long as the result of that diplomacy action didn't result in a battle, you could use your second move to make an actual attack.