I am trying to figure out the setup of the integration with an external system. When I go into the user account being used as the integration user account I see the name of the application in the login history. However, I can't see an app by that name listed in the Connected Apps, or in the App Manager.
If I go into "Connected Apps OAuth Usage" I can see that app listed, along with the users who are using it. But that's all the details I can get regarding the app. The "Install" button is to the far right side (not the "Uninstall"), which confuses me even more because I wouldn't think that an app which is being used still needs to be installed. And that's about all the information I can get about it. Is there another location to see the details of this app ?
I don't think you will be able to get details of app as those are created in an org on which you might don't have rights. One thing you can do is to use that app in sandbox login sandbox while using that app and after that you can install that app in your sandbox if you find anything un-usual then you can block that app and can revoke all access token.
The Same I did with Salesforce CLI
After this I was able to edit it's policies
Why we can't see the app?
For OAuth It's not necessary the app to be installed in your Org. You just need to authorise it very first time. Might this helps you to get a clear idea on OAuth
You want Create > Apps... , that's the menu tree for managing apps you develop.
The Install button only shows up for us under Manage > Connected Apps oAuth Usage when the app was installed and then uninstalled. If it shows up for an already installed app, it might be a different app. The app identity is not based on the display name you see on the list of apps.