Shape is automatically converted from "Shape" to "Shape-Path or Path" when I rotate

You can also right click and choose Free Transform Path. It works with selection and drawing tools active. When other tools are active like Brush tool, ensure that Shape layer is selected in the Layers panel then go to Edit > Free Transform path. Scale or rotate but do not distort, skew, perspective... because you will loose live shape capabilities which are corner options to adjust corner radius. 

• You get a warning, when you transform a live shape, like a rectangle. All that means is that it won't have the live shape, which has some nece editing features, without having to use transform. It's still a shape though.

What version of photoshop and Windows are you using?

Just using Ctrl+T to rotate a Shape should still keep it a shape.

Can you post screenshots or a video to demostrate what your seeing?

@defaultr6tz563kbsw8 

Instead of Ctrl+T, just put your cursor just outside a corner handle of the selected Shape. When you see the Rotate cursor, drag it. You can also type the amount in the Properties panel.

You can convert back to a Shape in Adobe Illustrator, which is much more robust for drawing than Photoshop (which is an excellent image editor.)

Jane