OK, I think I figured it out. While I have and use the Chrome browser on my PC I never signed into it. There was an option and I did just that using the account I created during the setup of the Nexus 7.
The strange thing is I went looking for how to sign out of Chrome on the PC and there wasn't an option. It will always tie the Chrome browser on the PC to some account once it has one to leech onto. Blood sucker.
I did find a strange way to sign out, though, and that was launching the Google dashboard, which in the upper right corner was the email addy (account ID) and it allowed me to sign out. The Other Bookmarks are still there in the PC's Chrome browser, but now the color settings icon is amber because I've deprived it of the ability to sync on the PC.
I do still have the Nexus' bookmarks local on the PC, which is what I wanted. I did see a video someone made when Google said you have to delete a user, but that isn't specific to the PC's Chrome browser only, or if it would delete the user from all of Google.