Tonight I looked at the nipple for the other toilet.
Taking lessons learned from the first one, I got a magnet and checked the pipe. It's not magnetic, it's chrome-plated brass.
But I had to take the valve off to have a look inside the pipe. Where is all my rust-colored junk coming from? Well, not from inside the nipple or valve, both were perfectly clean. Like the nipple, the valve was brass too.
It's a good thing the nipple is brass, obviously, but also because this 90 does not appear to be mounted to anything. Fortunately it was long enough to get a pipe wrench on it.
I noticed there was some blue silicone glue-like substance on the threads. White tape is there now.
So, on to other possible solutions for the rust issue. I'll try the bleach idea in the overflow tube on this one. It seems to have fixed the problem (a different problem) on the other one, but not enough time has really passed to say for sure.