I contend that it is abandoning the published instrument approach for a visual approach.
OK. I think your assumption that the aircraft forgoes the option of an instrument go-around merely by accepting that it can land on a different runway is verging on unreasonable. In my scenario, it may not even be able to see the airport at the moment of that exchange.
Also, it is poor ATC terminology, and serves to cause the type of confusion being discussed here.
I agree. That's really at the heart of the thread, though, isn't it? If ATC makes the type of clearance explicit, you
know what to do for the missed. Perhaps the lesson from this thread for ATC is to make the type of approach quite explicit(?).
Also, the OP clarified that he was indeed asking about a visual change of runway.
But a circling approach (and the sidestep you described)
is a visual procedure. If you don't get the appropriate visual reference by MDA, you can't continue with the circling/sidestep. If ATC could always wait until the aircraft was visual with the airport before issuing a visual approach to another runway, we'd have no need in general for circling approaches or sidesteps. You'd just wait until they'd broken out on the 13 approach and then clear them for a visual approach for 18. So the circling approach caters for a situation in which you know well in advance that you want the aircraft to land on a different runway.
BTW, can you give an example of a published sidestep, please? I don't think I've seen one.