So... as the motor cools down the battery mysteriously re-charges and then will spin the starter no problem!!, seriously, gentlemen !! Brushes and bushings in the starter are the culprit, I say.
It's no mystery.
Wet cell batteries do exhibit a self recovery phenomena. It's a chemical process in there. As the chemicals move around in there, fresh(er) acid gets to the plates. Certainly isn't a full recharge. But, it's gets better than previously flat.
It could also be brushes and bearings, I suppose. But, when mine went bad, it actually started better when warm, as there was less motor turning required (armature shorting to frame). Eventually, it didn't matter what temp it was.
Cheers,
Edit: In any case knowing the battery voltage is non-invasive and a logical first step to troubleshooting. Otherwise, just keep replacing parts till it works again.