I think the idea of calling a wide, extended, or whatever downwind is to provide more situational awareness about where you are, not to try and make yourself part of the circuit traffic. If you call "on a four mile downwind", you've decided that a position report is appropriate for some reason (due other traffic), you're advising that you're aligned with the downwind leg and you're about 4 miles out, you do this to give people useful information, not to try and assert yourself over someone joining via a 5 mile final or to get an aircraft on the ground to wait for you.