Hmmm could it be an RCS cluster on the LM? The LM is pointing at the lunar surface at this point, isn't it? ...Seems a pretty likely candidate, they stick out from the body of the LM, meaning you're not likely to see the rest of the LM, and it is definitely outside the window, but near the CSM.
Yes, I thought of that after posting here, then inspecting as many still photos as possible of the top of Eagle from inside the CM. It was after 3 am NZDT so my brain was fading, but I couldn't see anything near the top of the LM that looked like the ring in the video, and particularly not the VHF antenna at the top rear of Eagle. There are still the S-Band and Rendezvous Radar antennas to consider, but the solid and bright, shiny appearance of the "ring" in the video reminded me mostly of the RCS thrusters. Lighting conditions at the time could have made the view down the nozzle of a thruster look like a ring from that angle if very little light got inside the nozzle.
Edited: Having previously looked at the still photos and now viewing the video again, I can understand what we are seeing. The entire sequence runs form 0:12:27 to 0:13:13 and the most relevant part starts at 0:12:54 when we get a close-up view of an antenna on the left and a bunch of RCS thrusters near top centre.
The concentric rings that form the thrusters' nozzles become more obvious at 0:13:07 when the zoom is adjusted and we are viewing just the light reflecting off the two nozzles at right angles to each other (the "rods" in my first post), and in between them but a little higher than dead centre is a front-on (or top-down) view of the single thruster that is facing the camera (the "ring").
So, do those particular views match the "10" in the first video?