How do I tell if a symmetry is gauge or not?

I understand the interpretation given in topics like this one that gauge symmetries are “fake” in the sense that they do not represent an actual difference in physical states. I also know that gauge symmetries are those that are local. But how would I see the relation between the two? Or, to put it another way: given a Lagrangian, how can I see directly whether the “symmetry” is a real symmetry or not?

For concreteness, consider the following Lagrangians:

$$mathcal{L}_S = frac12 partial_mu vecphi cdot partial^mu vecphi – frac12 m^2 vecphi cdot vecphi $$

$$mathcal{L}_M = -frac14 F_{munu}F^{munu}$$

In the first case we have that $vec phi to R vecphi$ is a symmetry when $R in SO(3)$, in the second case we have the usual gauge symmetry of pure electromagnetism. Why is the latter a gauge symmetry and not the former? If we label states with the value of the fields, why do $|A_murangle$ and $|A_mu + partial_mu lambdarangle$ represent the same physical state but $|vecphirangle$ and $|Rvecphirangle$ do not?

I had a feeling this might have to do with the Noether currents, so I calculated them: in the first case we have three currents given by $J_a^mu = i partial^mu phi_a (T_a)_{bc} phi_b$ with $T_a$ the generators of $SO(3)$, in the second case I get a current of the form $J^mu = -F^{munu}partial_nu lambda$. Does it have to do with the fact that there are an infinite number of currents, one for each $lambda$?