For your examples you could craft a routine to tell you this information
anything ending in a gdb will be file geodatabase
do a describe on a sde connection and look at the instance will tell you that database it is based on the connection string e.g.
desc = arcpy.Describe(Database Connections\Localhost SQL.sde)
cp = desc.connectionProperties
print cp.instance
> sde:sqlserver:svr-gis-01
The only issue I see are the old style SDE connection where the instance will be a port number.
We're working on this! This status does not guarantee that the functionality will be in the next release, but development work has begun. Release cycles vary by product so make sure to check the product life cycle information to get an idea of when to expect the next release.
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