I think we argue too much over terms.
At one time these cans were at the top of the heap. They aren't any more. I have heard that there are two versions of the the HD 650, but in any case they sound great. But they are no longer the greatest and we can hear the difference of cans that are better. A reasonable person could call them "mid-fi" based either (or both) on the bases that they are no longer among the top tier cans on the market and those that are have superior sound (at least to most) than the HD 650/600.
This argument will continue unless and until we come up with an objective way to define what qualifies as "hi-fi" and will so qualify for many years to come. Ah--there's the rub. The 50's and early 60's definition of "hi-fi" now includes many cans priced below $50 street. So why are the best cans of 200x forever to be rated as "hi-fi"? Therefore it is logical that cans below the current best generally available to the public are not in the top tier. If that top tier is called "hi-fi" then those below it might be called "near hi-fi", "hi-fi lite" or (God forbid!) "mid-fi". I guess that we can call the new leaders "higher-fi" and more folks will be happy. Cans like the HD 600 & 650 are still very good, but they are no longer the best. Someday they may not even be very good. I've seen (and heard) that happen over my decades of listening to cans beginning in the mid-60's. Cans that were very good or excellent decades back are entry level or worse when compared with todays cans.
And so it goes with cans, cars, TV's and many other products.