Re: If starter disengages too early, is it a low battery or ?
I'd say your are correct---sort of. Yes, the starter is weak for some reason. Could very well be low battery, but of course it could also be poor connections or bad cables. Or even a starter problem.
As the motor flips over compression, (or fires) the flywheel "gets ahead" of the starter pinion and kicks it back down. If the starter were keeping pressure on the flywheel even over the compression kicks, it wouldn't be disengaging.