The 650 size wheel is a closer match to the 27" inch rim than the 700 rim, which is theoretically interchangeable with 29 inch wheels. Most higher end mountain bikes are now using the 650 wheel instead of the 29 inch / 700. The 650 wheel is supposed to give you the best of both worlds -- the crisp handling of the 26" wheel with the ability to roll over obstacles easier that you have in the 29" wheel size.
That's correct, but you're referring to 27.5" rims, a.k.a. 650b.
The 27" rims we've been discussing are slightly
larger than 700c (and 29") rims. They were standard on "10-speed" road bikes of the 60's and 70's like the Schwinn Varsity and Continental.
It's confusing because the names sound like dimensions but are effectively just names. They're derived from inconsitently imagined hypothetical (well, more like
mythical) dimensions and aren't aligned with reality. Luckily, there are specifications assigned to those names.
700C wheels have a bead-seat diameter of 622mm, for 27" it's 630mm. 8mm difference in diameter is 4mm radius, so if you put 700C wheels on a frame designed for 27" the brake needs to reach 4mm further -- less than 1/4 inch.
That's right, but sometimes...
Once upon a time, don't even remember what bike it was, I installed 700c wheels and found the brake pads were already at their lowest position but didn't reach. I had to swap out the calipers for ones with longer arms.
But yeah, it was more commonly like I had with my first "10-speed" bike, a Peugeot. It originally came with 27" steel clincher rims. I re-laced the hubs with 700c tubular rims and the MAFAC center-pull brakes just needed to be adjusted a little.