I was about to say you have already found the only two options (other than changing hand or doing something really wierd) but looking here
step 2
it looks like a useful variation is to use your second grip but with the handle pointing downwards. This means holding the blade in a different way, but might give better balance than if the handle points up, and improves visibility too.
If you check out shaving videos you will see some creative grip styles. There is no right or wrong, just what lets you get the blade where you want it while keeping good grip and control.
The best approach is to visualise where you want the blade to be, then work out where the handle works best (up, down, or even in line with the blade), then see how you need to hold the razor to achieve all that.
The good thing about the "standard" grip is that it's a known quantity; easy to learn, gives good grip, control and safety. But once you've learned it and gained confidence and a feel for things you can branch out into doing your own thing.