Everything you've said makes sense and is as expected - except this:
By the way with the bulb the voltage dropped to about 7V.
That seems a bit high - and, to my mind, isn't consistent with your other observations. If this number is correct, there may be other causes like the meter leads picking up stray EMF.
For this exercise, the 12V globe was OK, since we already expected there to be no real current capability. However, if you ever come up to a situation where you are really unsure, get yourself a 120V incandescent. The last thing you want to do is put 120V through a 12V globe.