I think it's this way because most traffic heads to L Street to get onto I-80 since there is not entrance on Q Street. I remember growing up in Old Millard around Norris and going through this area often. We always exited I-80 at Q Street but had to enter from L Street.
It's been decades since I've lived in this area, but at least back then, in the morning, this intersection needed two left turn lanes. Going straight was never a problem. So you wouldn't want to go with the first concept if that is still true today.
I do like your second idea. The main problem is westbound and eastbound wouldn't be able to go at the same time and that may make the time for a full cycle on the signal longer.
I lived in Omaha from 1984 to 1992. Back then, Q Street was only a 2-lane bridge over I-80. It had no exit from westbound I-80 until 1989-90. I suppose that could be another reason why 120th and Q is the way it is.