I don't understand the OPs approach. Doesn't the U-Turn require crossing lanes. Of course if the cross street is much smaller and perhaps only has one lane in each direction, then there is no crossing.
I realize the OP's questions is what is this called rather than a question on best practices. But I'll chime in on that. I rarely ride where I need to cross multiple lanes with heavy traffic. So, I never have trouble moving over into left turn lanes. If I did need to, I'd probably stay right, cross, then stop and orient myself to cross again on the green, What some here are calling the "Swedish Turn". I've heard this called a "box turn".
Heck even when driving, trying to move from the far right to the left in heavy traffic is hard enough. Most of us would make the needed lane changes well in advance by taking advantage of gaps that open when the cars are at max speed between intersections. Around where I live, heavy traffic moves too fast for a cyclist to do this. When the traffic slows for a red light, it's bumper to bumper and no one in a car or on a bike is making any lane changes.