::I can't address all of your points, but for what it's worth, Larson's Canute the Great calls him Sweyn, and simply indicates that he is son of Canute (i.e. "Canute . . . selected his illegitimate son Sweyn", and having thus identified him, just Sweyn thereafter).  It indicates that at different times he was (probably) regent of Denmark and then regent of Norway, not king. This work is from 1912, though. The problem is the typical one with Google Books - nothing recent enough to show current usage is available due to copyright.