Since you are using Rename class wizard, this acts as renaming the same class and not moving to a new class.
So moving partial rules (only properties) is not a scenario. If you notice closely the screen we get to optionally select rules is the one where rules reference this class name in some way.
That means we don't get an option to move selected instances in that class. If the class in question has 4 properties, 4 data transforms, 4 activities and so on...
All of these rules will be refactored to refer to new class and these are not optional/selectable. There is no way you can only select few rules here.
However there can be a scenario where some rules refer to this class name in java step or some property (other than applies to class), let's say we have a data transform at framework level for DCR which refers to this class. These kind of instances come as selectable.