You can easily tell that this motor is very high current. The design of the brushes
(large copper pigtails and a big spring) indicates this. A motor that size can be
designed for intermittent duty at very high currents (for instance electric screwdriver
motors).
You must measure the current to see what the motor needs to run no-load. I suspect
this is anywhere from 0.5 to 2A and the full load current is in 5 to 15A range. You need
extra current to start from stationary as the static friction of the brushes/commutators is
significant for a high current motor. Static friction is often significantly more than dynamic
friction which is why once you you've started it with your fingers it stays running.
If your 12V lead acid battery is good and had trouble then it could be the motor, but
it could be that you used inadequate wires or connectors.
99% of problems people have with motors is not realizing how much current they need,
prove that you've measured things and provide some data to the contrary and we'll
be more likely to entertain the 1% hypothesis.
I own perhaps 50 motors of all sorts and sizes, trust me, you don't have enough current!
Im sort of coming to the conclusion that by size you mean physical size.
If so, its irrelevant almost to the point of being meaningless , both for the battery and motor.
Knowing the weight might be useful though for some applications.