Perhaps take the machine on which you assume FMS is NOT installed off the network. Then access the admin console from a browser on the other machine, the one with FMS installed. Do you get errors? If all goes well then the chances are you do have FMS installed on both machines. When the second machine is on the same network each will detect another installation. FMS is not the same as an app. I'm not sure how to uninstall it but it isn't as simple as dragging an app to the trash.
Hi Rick -
At risk of showing my Luddite-ness, I can't figure out how to reply, so I'm posting this as "another answer. Your proposed solution was BRILLIANT!! You are now my new best friend (sorry about that!).
That fixed my issue as well. My test server was still running on my laptop. I quit that, then restarted the FileMaker Server 13. Now all is well.