one of the stems has budded. My question is, just what kind of Ivy is this
or better yet is it even an Ivy? Also, can I grow it indoors under the 150W
incandescent bulb without some special treatment?

This time you have Hedera helix, English ivy. You can probably grow it indoors,
but it will want a little more light than the pothos, yet it wants to be
cooler. It will be better off on a cool windowsill. Also, especially since it
came from outdoors, English ivy is susceptible to spider mites and many other
ills. Keep a close watch on it.

Iris,
Central NY, Zone 5a, Sunset Zone 40
"If we see light at the end of the tunnel, It's the light of the oncoming
train."
Robert Lowell (1917-1977)