I think it does have to do with the move, but it says, "x avoided the attack" when an attack that is intended to hit multiple targets (like in a double battle) misses. For example, in my HG game, it says, "x avoided the attack" when I use Blizzard, which hits both targets in a double battle. Whereas, when an attack like Thunder misses, it just says, "x's attack missed" because Thunder is only meant to hit one target in a double battle. Maybe this is it? I don't know.It is determined by the move. For instance, Razor Leaf is always evaded, whereas Tackle always misses. I guess that some attacks (think Earthquake) would sound stupid to 'miss', so they have to be 'evaded'.
And for the record, I'm using Blizzard instead of Ice Beam for the in-game story mode because I'm too lazy to play that Voltorb flip game.