Thanks! I still don’t really get it, but telling me I need the Lagrangian formalism is helpful, and tells me what I need to study in order to really understand Noether's theorems.
I’ve heard of the Lagrangian before, but my knowledge of physics is probably somewhere around the level of someone who has only just finished the first year of their degree, at least judging by the modules my alma mater lists for their physics degree (I got Software Engineering from them 14 years ago, all my physics knowledge since then is personal geekery).
To get a feel for it (and why it may not necessarily be intuitive) consider the following pairs of symmetries and conservation laws:
- The laws of physics are invariant under time translation (i.e. repeating an experiment at different times gives you the same result ceteris paribus). The corresponding conserved quantity is energy.
- The laws of physics are invariant under spatial translation (i.e. repeating an experiment at two different places gives you the same result ceteris paribus). The corresponding conserved quantity is momentum (this is a vector: one component of momentum for each possible direction of translation).
- The laws of physics are invariant under rotation (i.e. repeating an experiment under different orientations gives you the same result ceteris paribus). The corresponding conserved quantity is angular momentum (again a vector, since the rotation group SO(3) has three generators).
- Electromagnetism, constructed as a classical field, exhibits an internal symmetry in the field quantities. The corresponding conserved quantity is the electrostatic charge (actually a 4-vector current).
It is not a priori obvious from from the kind of arguments you've supplied why these should be the case. Demonstrating these would require access to the Lagrangian formalism (i.e. action principles) and how it behaves under these symmetry operations. I would say that, as you suspect, you are fundamentally misunderstanding the situation.