I have some confusion over what exactly constitutes a symmetry when trying to apply Noether's theorem. I have heard both that a symmetry in the action gives a conserved quantity, and that a symmetry in the Lagrangian gives a conserved quantity.
Both of these statements are confusing to me. From my understanding, a transformation is a symmetry of the action when $\delta S = 0$. However, on the classical path, I would expect this to be trivially true for any transformation, not just a few special ones, since the classical path by definition minimizes the action. Should I actually take $\delta S = 0$ to indicate a symmetry only if it is true for all paths, and not just the classical path? If so, keeping in mind the conserved quantity which this corresponds to is actually only conserved along the classical path, it seems weird that I would need to know something about the action on all paths in order to make a statement about a conserved quantity on the classical path. It seems like you should only need to know information about the action on the paths near the classical path in order to make a statement about conserved quantities on the classical path. Is this incorrect?
The statement that a symmetry of the Lagrangian gives a conserved quantity seems strange because there are transformations which give conserved quantities that do not leave the Lagrangian unchanged. For example this is the case for any transformation which changes the Lagrangian by a total derivative. It seems weird to call this a symmetry of the Lagrangian if the Lagrangian actually changes. Am I misunderstanding this terminology?