88 gt starter

Don't confuse starter relay with starter bendix. The starter relay is on driver side inner fender. The Bendix is part of the starter motor, it is a solenoid.

The Ford starter solenoid on the inner fender is fine if the starter spins and stops with the ignition key. The solenoid is likely to either not work at all, or to keep working with welded points, and low voltage seem to be the culprit in both cases.
The mini starters, GM and most everyone else have the relay and electromagnet for the Bendix throw out built into one unit on the starter top. Blown88GT wanted to make sure you were not just a Chevy guy.

The sound of the first attempt is of the starter spinning without engaging the flywheel or flex plate. The second attempt noise is of the spinning starter briefly hitting the flywheel or flex plate. That’s hard on teeth!

So, the starter needs replaced or rebuilt, and the teeth on the flywheel really need inspected before putting the new starter on.