Sounds like the game you called boxball was similar to what we called slapball. Open fisted with pitching. It could be played in parks or the street.

We also called it triangle when there were only 3 bases, 1st third and home, and it was played along the width of the street.

Punchball didn’t include pitching. You’d throw it up and hit by yourself – overhand side-armed or underhand. You could really whack those balls far.