Good questions, David.
I didn't know there was a light in the ashtray.
After more fiddling, I believe I am getting power to the lighter. The lighter does seem to get hot when I turn the key to position 1 and manually hold the lighter at the deeper, "heating" position.
Still, however, the lighter will not stay in and heat on its own. I'm left thinking either that I got the wrong part or that the lighter socket is broken, perhaps through prolonged use of lighter-powered cell phone by a previous owner.
If anyone can confirm the proper part number, I'd appreciate it.
'92 325is (sold)
'72 250C (sold)
'71 280SE Euro (totalled)