The simple answer is none of them is correct. But, here's the thing that get people confused, even though it's not correct, but you can still use the 'different vanishing points' technique in some limited cases. I guess a lot of people get confused with vanishing points. Vanishing points exist first, thus they affect the objects in the scene, but it's not the other way around. The position of the vanishing points cannot change just because the objects in the scene move or rotate. Even though there's no object in the scene, vanishing points will still exist. You see, the reason vanishing points themselves exist in the first place is because there's a camera shooting the scene, if there's no camera then there's no vanishing points. Thus the only thing that can change the position of vanishing points are when you move or rotate the camera (changing the camera position or angle), while the 'distance' between vanishing points can change when camera lens' focal length also change (you may have heard the term 50mm or 200mm in camera lens, that's the focal length). So in reality, there cannot be two different sets of vanishing points in one scene, leave alone different 'distance' between the vanishing points. There can only be one set of vanishing points and one distance in each scene, no matter how many objects you have in that scene. As many have pointed out below, the confusion starts when we can actually try to change the position of vanishing points, while maintaining the same 'distance', to portray objects rotation relative to the camera. There are some cases where you can successfully and correctly do that, but in some other cases it doesn't work. I've attached some 3D reference I made in Blender to show that, in that particular scene and camera angle, when the object rotation is counter clockwise, it does work, but when the rotation is clockwise, it just doesn't work.

Yes, of course, that means the object is rotated.

I’m pretty sure the distance between vanishing points should always stay the same. There’s a gif on Draw a Box that explains this https://drawabox.com/lesson/1/17/rotation (it’s a rotating box, where you can see the vanishing points moving with the box). I think the distance needs to remain the same or else it would be like using different camera lenses. For example, if you take a picture using a wider lens (the objects look more distorted), take a picture using a long lens and try to put them together, it’s very obvious there not part of the same picture (moderndayjames explains lenses briefly in one of his videos: https://www.youtube.com/watch?v=2XF5YuAK63I , about 4 minutes into the video). I tried to illustrate this using your example number 2, but more extreme. In my example, the green boxes look like they could be in the same scene and the blue boxes look like they could be in a different scene together. But together, the green and blue boxes have very different levels of distortions (because the vanishing points are not the same distance apart) so they don’t work together and make things look weird. Basically the lens stays the same, so the distance between the vanishing points stay the same. Hope this will be of some help! (ps I’m not sure if this is the best way to explain it, or if my explanation is 100% correct, but I’ve heard a lot of people who have much more experience talk about how the distance between vanishing points stays the same in a scene)

Both are wrong, that's not how perspective works. Boxes with the same orientation will share the same Vps. Remember, parallel lines converge at the same Vanishing point in perspective. If you are trying to rotate a box, then No the length for the VPs do not stay the same for each rotated box. I suggest looking at Scott Robertson "How to Draw" page 25. The two vanishing points for a box should meet at 90 degrees on the Station Point. You rotate that 90 from the station point to rotate a box. The VPs for the rotated box will vary in length due to basic common sense geometry. You'll get it when you see it. hope this helps.