The latter...your legs still get blood circulation while you strangle.
asphixiation most likely.
Blood carries oxygen throughout the body so either way you are deprived of oxygen.
I think you would die without air before the circulation being blocked killing you. Blocked circulation would cause a stoke (i think) and they would definitely be able to tell the deference in the autopsy if that was the case.
Lack of air cause you can survive a stroke.