Keywords modified in Lightroom CC do not sync back to Lightroom Classic. This is a known limitation, and there don't appear to be plans to resolve it.
That's disappointing. But what makes it especially frustrating is that the behavior is unexpected and thus reads like a bug. In my case, I spent an hour troubleshooting synchronization between Lightroom CC and Lightroom Classic before looking online, and discovering that it was "by design".
Given that, at minimum, it'd be nice to see a warning given when users attempt to add or modify keywords in Lightroom CC for photos that are synchronized with Lightroom Classic. That way, they don't make the mistake of updating their keywords in Lightroom CC in anticipation that they will be available later in Lightroom Classic. This could even be a one-time warning.
The goal is to ensure that the user is notified of the limitation since it's not otherwise intuitive or expected.