Is the Base URL in the Recorder/Test Case Used for Test Context?
No, the Base URL is only used if the test you're recording is outside the Salesforce context (e.g. you're recording a test that starts in google.com). In that case, the Base URL is used to land into that 3rd party site.
In the Salesforce context, Copado will rely on the user's (through the credential) configuration to determine the landing page. Alternatively, within your test, you can use the Open command to specify a URL you want the test to land in.
The recommendation is always to record the navigation where you want the user to land, so regardless of which user is being impersonated (credential), you consistently land on the same tab (e.g. open the App Launcher and click on the Cases tab).