Based on code inspection it seems that Chrome (not shipped) and Safari perform browsing context name resetting based on whether the top-level browsing context has an opener, not on whether it is auxiliary. These are distinct as a non-auxiliary top-level browsing context can still get an opener by being name targeted.

Nika told me that Firefox does have a concept of an initial opener so there might well be some valid uses of auxiliary browsing context.

It seems the one other use of this distinction is #313 due to "familiar with".