If the starter motor spins, but doesn't catch and crank the engine--sounds like the issue is related more to the starter motor system itself, not necessarily the charging system at all.
I agree that battery voltages should be checked while under starting load, in addition to while running, as well as off and at rest. Rule out one whole potential problem at a time before fixing what isn't necessarily broken.
If the DC voltage drops down past 10 volts or so when actually cranking the engine, that's a problem. If it stays above that, your battery is probably fine. A load test at your local battery store will tell more of that story than this test, but it's a start.
Also, another poor man's test of the charging system (if you don't have a voltmeter) is to see if the headlight gets brighter as the engine is revved, returning to a bit dimmer at idle.
My motto: "If it ain't broke, fix it 'till it is".

All the best,
Shane