Introduction to Gauge Symmetries: Good, Bad or Ugly?

I'm trying to come up with a good (as in intuitive and not 'too wrong') definition of a gauge symmetry.

This is what I have right now:

A dynamical symmetry is a (differentiable) group of transformations that respects system dynamics, ie maps solutions to solutions.

A rigid symmetry is a dynamical symmetry that maps solutions to different solutions. A rigid symmetry has a Noether charge that is only conserved on-shell, ie dependent on the equations of motion.

A gauge symmetry is a dynamical symmetry that maps solutions to identical solutions up to 'parametrization' or 'gauge'; in particular, the solutions correspond to the same initial conditions and physics and only differ in their mathematical description. A gauge symmetry has a Noether charge that is conserved off-shell, ie independent of the equations of motion.

As an example, we take classical mechanics: In general, time dependence of the solutions matter as reparametrization changes velocities. However, in the relativistic case 4-velocities are constrained to 'length' $c$ and dynamics need to be independent of the particular choice of the 'unphysical' 3-velocities.

First of all, is this correct? If so, is there a better choice of wording? Should anything be added?