The text applies only to a situation where the BR overruns first base, period. That's clear since it starts out, "A batter-runner who overruns first..." It doesn't apply to when a BR rounds first.So which one of those does the text from the rule below support?
"A batter-runner who overruns first and does not without delay attempt to
advance to second is committed to return to first and stop. The runner, off base, may not stand motionless"
What you're doing is taking the look-back rule and applying it to a BR who advances beyond first base and then stops. Yes, a BR who goes beyond first either by overrunning it or rounding it and then stopping can end up violating the look-back rule. But the similarities end there. It doesn't mean that overrunning and rounding are the same thing.