What is to be seen in your gif is the behaviour of my original post, although with the axes flipped.
Thank you for adding grid and lines, they might help describe my problem:
This is exactly what I meant:
This is what I am trying to get rid of:
As you can see, the sphere rotates around the yellow x-axis.
What it should rotate around is the not yet drawn stationary “x-axis” that lies on the grid.
Just as the y-rotation in your sketch always happens around a not drawn y-axis on the grid.
This is not an assignment and I am not trying to bait you into doing it for me 
This is a random sketch idea I had (further problems here lol P5js: 3D Typography Problems/Bugs?) where something I thought would be trivial apparently turned out to be kind of tricky and I’m almost certain there’s some clever math involved in solving this.