OK, now when you say you want to move on the XY plane you do understand that Y in godot is up/down right? Or did you mean the XZ plane instead?
Also just so we are on the same page Sin and Cos are essentially alternating periods of the same thing:
Image courtesy of 1ucasvb's lab. Check out this post for further details. The whole blog is worth checking out. Also while I'm throwing useful calculus resources around heres a playlist of 3blue1brown called the essence of calculus. The good stuff™. Also essence of linear algebra.
Normal, Tangent and Bi-normal have to change, you will not escape that, but you can do everything you need to modify the vertex positions first and only manually calculate normal components last when using
And learning to dissect the code of others is a very valuable thing to-do. ;)
Aw heck here is a page with a gif showing also the tangent:
edit nr bazilionth. Also before I forget: from the texture blending/flaky lighting thread I noticed in the scene you uploaded that your normalmaps were oriented wrong, so here is a normalmap I have baked for using in godot that shows which way the axis gradients should be oriented in:
red highlights from left, green highlights from top. In terms of extrusions from the surface.