Retro steps:
1. Change gamemode to battle ("/battle" command) -> name becomes white
2. Change gamemode to ctf ("/ctf" command)

Expected result:
The color of my name updates according to the team that I belong to (blue or red)

Actual result:
Name stays white until I get the flag.

Video:
https://streamable.com/boglnl

Thanks for the report! This has the feel of something that should be easy to fix...

this happens vice versa when switching from ctf to battle too

From what I've been able to tell so far, this happens specifically when using a 6.4 client to join servers running 6.3 or below.

Does this fix the problem?

Somebody contacted me offsite to say that the above build isn't always working for them. At this point I don't know how many hours I've spent staring at the older name code, trying to figure out what's wrong with it, but my tentative conclusion is that this is not a new problem. If you join a 6.3 CTF server as a 6.3 client, and switch the gamemode to battle, your name colors won't always be right there either. I haven't given up yet, but it's possible there's may be no way to force 6.3 (and below) servers to behave other than for people to stop hosting them.