That's like saying 'it's battle magic because it's got battle magic mechanics'. It doesn't really tell us why something is or isn't battle magic.
Here's what I mean by the math not adding up. If wind is being used as a form of energy, the amount of wind needed to do something should correspond to the energy output of that something.
If we consider fire spells, it should take a whole lot less wind to boil a cup of water than it does to create a huge fireball, because the huge fireball spell requires far more energy input to match its energy output.
Most battle spells seem to do big, energetic things, or less energetic things over and over again to many targets in a way that the energy requirements and complexity would add up. Invisibility on one person is a regular spell, invisibility on 40 people would be a battle spell.
The force required to lift someone into the air and move them less than a mile isn't really all that much compared to the scale of most of the stuff that battle magic seems to do. That's what makes me think that a writer put that spell on the battle magic list because it has a battlefield use and so they thought 'oh hey, battle magic' and not because it being battle magic makes any sense in the world's physics.
There should probably be plenty of less complicated and energetic spells with battlefield uses that aren't battle spells, and plenty of 'battle spells' that have nothing to do with warfare.