@LorenDB Yes, I think you are (almost) on the right track. In your normal pictures, notice that the neck is sitting flush on the base of the thumb. In your “rotated” you are opening up a gap between the base of the thumb and the neck. That gap is moving in the right direction, but I think how you are getting that gap could perhaps use a little tweak. When I simulate your normal position, I feel really tight and have a hard time doing what you are trying to do. Oddly enough, in the rotated pic, it looks like there is even more tension. What we want to do is drop the palm and the base of the thumb away from the neck AND do this in a way that releases tension. Try this: go to your normal position and fret the 5th string second fret. If I am looking at it correctly, your wrist will be bent backwards and there will be little to no free space between your palm and the neck. Now, relax your wrist and let it drop down and to the left, in the natural plane that wrist bends. If you do this correctly, this should be a totally relaxed position. When I do this, I have only three points of contact: the pad of my thumb, the top of my pointer finger (top/inside of pad) and the base of my pointer finger (about at the crease where the finger meets the hand). When I do this, I have a sizeable gap between my palm and the back of the neck. I can put my normal size computer mouse in there enough to hold it. It is also feels totally relaxed. I could sit like this for 30 minutes. I didn’t actually test that, but it seems like I could. This should be a more relaxed and capable default position. I think you can also tweak it a bit better by also dropping the elbow and arm a bit. And to your item of concern, I hope you can fret the fourth much easier. Let us know if that is making sense. If it doesn’t and you want to facetime or something similar, IM me. We can also get videos going on this thread if you prefer.
I think @Zachary makes a great point about the fingers pointing nearly perpendicular to the fretboard. It is generally easier to angle them so that they point at your right shoulder or ear. I think if you drop the wrist and elbow, that will help rotate things to where the finger pointing direction is much closer to ideal.
One last note… there are a few times I collapse my wrist up into the neck, most notably when doing something like doing an F chord with a wrapped thumb. However, when I do an F barre chord, the opposite happens, the wrist drops, the thumb moves to the back of the neck and a huge space is created between the neck and my palm. My point being, there are times for different thumb positions and wrist angles, so I don’t want to create a “rule” where one NEVER has a wrist that collapses up into the neck. It’s just not ideal for most playing.