I think it has to do with the movement and the function of the movement. I've heard more than once an instructor that calls it "watershed" describing it as an actual watershed; like a slanted roof shielding your head from rain. The slant in the parry/watershed/whatever deflects a downward attack off and away from your center as you enter to strike. The energy from the attack deflecting off the watershed is also redirected into your own cut to power your counter attack.
You know, the term "watershed" rings a bell. Not sure where I heard it..
From what you are describing, it sounds like a derivative of Kashima Shinto Ryu nagashi,
The youtube video you posted actually looks like a rising cut to the wrist, but that's not neccessarily a different movement.