First of all there was some discussion about this some time ago where a fellow mentioned the name "Collar tie." And Ed Levin did some FEA research and found that the "collar" is only acting as a "tie" (in tension) when it was low between the rafters. Somewhere around one foot above the rafter foot in elevation or lower. Above this point that "collar" is in compression, that is holding the rafter apart or from sagging in.
This fellow mentioned that the name really shouldn't be "collar tie" but should be "collar beam" as it really isn't tying, again unless located where mentioned above.
He did however have a third name which escapes me at the moment. These three names would be used for a "collar" depending on their actual location in the frame, based on elevation above the rafter's foot location.
Since reading this discussion, again some time ago, I have tried my best to use the term "collar beam" to describe the beam going between two rafters.
You are correct that there could be these three locations, and I feel that the correct location depends on several factors. Some of these would/could be length of rafter, roof load, rafter pitch, over all frame design, height clearance needed (head room).
It maybe that each individual frame design has to be carefully analyzed to determine exactly where the collar should be placed.
So much so that you can't just say, one location is the right one and the other two are wrong....
A common answer is "it depends"......
