Yea, Proxima Centauri is the red dwarf. I'm not sure about Alpha Centauri B's color. Star color is weird in and of itself; while we call stars like Alpha Centauri A and the Sun "yellow dwarfs", they're actually both white - the Sun only looks yellow because the atmosphere scatters more light in the blue end of the spectrum than the red end. Were you standing on the moon and staring at the Sun, it wouldn't appear yellow at all, though staring at the Sun usually isn't a good idea.

Incidentally, Proxima Centauri has an earth-sized planet around it that's on the inner edge of the habitable zone, though probably isn't actually habitable for the reasons OP pointed out.

There is a very tentative candidate for Alpha Centauri A. If confirmed, it would be a Neptune-sized planet also in the habitable zone (though barely), though again I must emphasize that this is *extremely* tentative. These candidates pop up around stars from time to time and a lot of them turn out to just be artifacts in the data.

If it is real, though, it should be big enough, far enough away from Alpha Centauri A, and close enough to our own solar system that we might be able to observe it with the JWST.

That's a long time off, though.