Your suspicion is correct, a charge is defined by that higher than the open circuit voltage of the battery, (It has to be to overcome the internal resistance of the battery) thus yours isn't charging!

You want to be seeing 13V+ WITH the engine running

You have a problem with the rectification side on the alternator, there are diodes (Think of them as electrical one way valves) that serve two purposes on an alternator.

A) The correction of AC to DC voltage needed by the car (Hence where the name derives from)
B) Allow current to flow in the one and only correct direction.

What is happening is that part isn't working correctly so with the ignition off current is going from the battery via the alternator and lighting the bulb. Where as normally the field excitation wire on the alternator acts as an earth when working correctly when the alternator is stationary.

Now when you turn the ignition off it's the reverse, the ignition feed is no longer and the field excitation wire has + potential thus why your light illuminates

When working correctly the bulb has two + feeds one for switched ignition feed and one from the alternator which is what extinguishes the light, indicating as is well.

Martin