Are you adding trim molding?

I like color on the ceiling, esp if it's vaulted.
So I would paint manchester tan on all but the accent wall and continue it up around the angled part of the ceiling on all sides. Then paint the accent wall the greenbrier beige and paint the flat part of the ceiling the same color. It would look great if you have molding to break up the areas painted in the simply white to offset the areas.

That may be too much for you so you can just paint the simply white on the flat and angled portions of the ceiling and then paint the walls manchester tan with the greenbriar beige accent.

I'd suggest going to houzz.com and searching on tray ceilings so you can get some ideas of how they are treated with paint.

Thanks for your help! I'd been searching "vaulted ceiling" on Houzz for ideas without much luck... now that I tried searching "tray ceiling" I'm getting a better sense of what it would look like.

I have a ceiling very much like yours. I painted the angled part the ceiling color. However, for me the angled part doesn't extend around the whole ceiling (just one or two walls, depending on the room). Also, I did not use white on the ceiling--I used a color a shade or two lighter than the walls. (I did use Simply White for the trim.)

I've often wondered if white ceilings are a regional thing--I don't see it much where I live (in California).

Typically( at least around here) all the angled parts are painted the wall color and only the flat part is the ceiling. So called "decorators" might have a different opinion