Who decides when "Roman Catholic" applies?

Get Religion offers a helpful thought experiment to capture how misguided some of the media coverage of "Roman Catholic" groups can be:

Let’s say that a bunch of retired journalists from the Los Angeles Times got together and, with a few converts who yearn for the good old journalism days in that great city, form a news organization that we will call, oh, the Society of St. Otis Chandler. This group rents itself some printing presses and, using a template of a vintage masthead of the Los Angeles Times in 1965 or so, start publishing a newspaper that they call — wait for it — the Los Angeles Times.

This makes some people confused, especially when the leaders of this new-old Los Angeles Times start making pronouncements that directly contradict those made by the leaders of the real Los Angeles Times.

Is everyone following this? Good. Hang on.

Now, the leaders of the actual Times clearly have the right — like it or not — to say who works for the real Times and who is aligned with this pretend Times. So how would these editors feel if major news operations kept writing stories about statements by the Society of St. Otis Chandler and calling its members Los Angeles Times journalists in good standing?

Now, unfortunately, there is one more complication. Suppose that some of these splinter Times people decide that the leadership of the Society of St. Otis Chandler have not gone far enough. Suppose that they start yet another group, one that claims that the leaders of the new-new Los Angeles Times are not only wrong on key issues, but that they are not even journalists in the first place.

Now, do you think mainstream journalists would go so far as to say that these people, the members of the splinter group that left the larger splinter Times, are, in fact, Los Angeles Times journalists?