iCloud Keychain will not update with new site password

Try running Keychains Repair Tool by opening Keychain from you Utilities folder, then from the menu bar choose Keychain Access, then Keychain First Aid, then enter your password, verify then start it. If there are any problems click Repair then Start.

If that doesn't work then try Resetting your Keychain. Via Apple's Instructions you just need to do these steps:

1. Open Keychain Access, which is in the Utilities folder within the Applications folder.
2. From the Keychain Access menu, choose Preferences.
3. Click General, then click Reset My Default Keychain.
4. Authenticate with your account login password.
5. Quit Keychain Access then Restart your computer.

There is no "Reset My Default Keychain" under Keychain Access Menu -> Preferences.

I am running macOS 10.13.1 (17B48)

Did you open the Preferences< Open Keychain Access, the go command - , (comma).