but a transom means merely a cross-beam, and from thence gets taken to apply to many individual features
by association (boat transom, transom window, etc)

I worked in architecture, in a former life, and a transome (yes, there was an 'e' on the end for some reason) was the horizontal section of wood which bisected a window's sash. The vertical equivalent is a mullion. I've never heard of a transom window but I can guess that one above a door is so called for the divide between the window and door frames.

Correct me if I'm wrong. I'd rather NOT look it up.