The differences between -async , physically_exclusive , logically_exclusive for command set_clock_groups during crosstalk analysis are the following:
physically_exclusive:
Means Timing paths between these clock domains are false, but only one clock can exist in the design at the same time. ETS/Tempus will filter out the SI interactions of nets/paths between these groups.
To phrase it differently, if the clocks are exclusive, then there should be no SI victim/aggressor interaction at all between nets clocked by physically excluded clocks.
asynchronous
If two clocks are asynchronous, it means that they don't have any phase relationship among them at all. So instead of using definite timing windows based on arrival times/skew etc, the tool will use infinite timing windows when calculating aggressors and victims, therefore you will see maximum SI impact.
logically_exclusive
Logically exclusive means the timing paths between these clock domains are false, but both clocks can exist in the design at the same time, so SI interactions between paths in these domains should still be considered. However crosstalk analysis will be done with regular timing windows based on arrival times/skew etc.