How should I address this window leak?

Where exactly are you? By your name I think I remember western WA area at one time?

DrewE, do you mean that I should disassemble the sliding pane, and replace old adhesive or butyl that will be hidden behind the rubber?

This is what you should use
Still I would double check it.
Usual installation is that glass is glued (with stuff like above) to the frame inside and then rubber/plastic trim is just extra security/cosmetic stuff.
I would pull everything off and reseal from clean.
The stuff above take several days to dry to rubber-like compound, so patience, patience.

It is a sliding window, and water is getting into the groove the window slides in, and also running down the interior wall, all in that corner area. If course, where we see water is not always where it is getting in :).

The weep hole is clear on the exterior, and is kind of huge -- a slot about 1.25" long. But I have not checked whether it is blocked on the inside. I can do that.

I am glad to hear black silicone will stick to the rubber, if it comes to that.

I would, and have, masked that area off and used soem black slicone to fill it in. Worked great and lasted for all the years we had it.

Is this a sliding window, or otherwise openable? Is the weep hole clear?

In general, the rubber surround is not expected to hermetically seal an RV window. Any water that sneaks past it is intended to drain through the weep hole; that's what the weep hole is there for. (Sometimes in fixed windows they have no weep holes, but even there the actual water sealing is typically the adhesive holding the glass against the metal frame on the inside, and the rubber surround is for appearance and some physical support. It's a little like a car windshield in how it's sealed and mounted.)

Well, the water does come in, and I am pretty sure that is where.

Last couple weeks, I reseated the window in fresh butyl and made a replacement weephole cover because I thought those were the problems. And maybe they were, but water still gets in, so this is my current suspect.

Glass is loose in the rubber.

Will black silicone stick to rubber?

Not leaking but for your peace of mind, some sealant in black would fill the crevice.

We have a window (dual pane) that the rubber on the curve has done the same thing, but there is no air leak. The window itself still closes off the area. Are you sure it's leaking air. I would suggest you talk to someone at an automotive/windshield shop. Most RV dealers have those companies do glass replacement for them on the rv's. They should be able to advise you. Take a picture with you to show them what you are talking about.