The way it was taught to us in Oregon & California, was that 'assault' occurs when there is a credible or actual attempt to inflict bodily harm: i.e, throwing a punch or kick whether or not it connects.
Battery, OTOH, is actual touching, as in the punch, slap, kick, etc actually makes physical contact with another...whether that was the intended target or not.
One can assault one person; fail to connect, and also be guilty of battery by accidentally hitting a third party.
Example: A swings at B; B ducks, and C, standing nearby is struck A. Assault upon B; battery upon C.
Good explanation. In Texas, though, assault could be verbal as well as a physical effort.