It's always been believed that if an aircraft (US registered) is
rented out, a 100hr check is required.

People believe all sorts of things that aren't true, including this.

I am now trying to find a reference to this in the FARs.

You will be looking a long time.

The above text doesn't seem to apply to plain rental. It describes
carrying a person for hire (implying that the passenger is paying)
and
offering flight training in the said aircraft.

This is correct. Those are the situation that require a 100-hour.
Rental does not.

Michael