Correct.  That would constitute two "battles".
 
(Thematically, it could just be assumed that the home land areas have some paltry defenses, not enough to do much damage, but enough to slow your armies marches).

It is definitely still considered a battle.  Anytime you move into an enemy-controlled area, it's a battle.  This is even true if you move into an enemy area that doesn't have any units (like an empty stronghold or an empty home area).
 
Technically speaking, you'd still draw cards for all your units, too (although in most cases, it doesn't do much.  However, units like the Knights and Necromancers can make use of those cards).  It does count against the 1-battle-per-Mobilize even if there are no enemy units there.
 
 
One note, however - if you move into an area that contains previously routed units, those units are NOT part of any battle that may happen there.  If the defender loses the battle, those units will be destroyed, but unless you stand those units up before the battle starts, they can't participate in any way.  (Units that are routed DURING the battle are still part of the battle, in case you have a card like Rally Cry; but Rally Cry won't let units that were previously routed join the battle after it's already started.  You could play the card right at the start of battle and they would join in, but if you wait beyond that, it'll be too late).