Plex server won't connect with one login - but it will with another

It means your server is using account #2
Therefore only clients which are also signed in with #2 are allowed to access it.

Sign in with #2 in the web app
Settings - Server - General - “Sign Out”
Then sign out the web app as well
Sign in to #1
You will get a message about an unclaimed server. Claim it now.