Yeah, I'm pretty sure it's an artifact. I'm seeing the same drift in the cylinder geometry, which definitely shouldn't be happening. A cylinder has the same internal geometry as a plane, so turning while moving at a constant rate should give you the transformed version of a circle.

Also, I notice that doing the same thing in the Klein model gives you very weird curves. Maybe you don't have the turning logic quite right? It should turn at a constant rate in the internal geometry, not in the plane geometry. (That shouldn't matter for the hyperbolic disk because it's conformal, so I guess something else is going on.)