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    $\begingroup$ I'm not sure what exactly the question is asking - it seems you want to know what the difference between a gauge symmetry and a "normal" symmetry is? What about the definitions you have seenwas not satisfactory? For questions on the nature of gauge symmetries see: physics.stackexchange.com/q/266992/50583, physics.stackexchange.com/q/126978/50583 and their linked questions $\endgroup$
    – ACuriousMind ♦
    Commented Jan 6, 2023 at 22:36
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    $\begingroup$ I think the question is indeed justified because some texts just say that gauge symmetries are unphysical, or redundancies, without extra qualification, while in fact there are indeed extra implicit assumptions to make this statement correct. The judge of whether a gauge transformation is a true physical symmetry or not is whether it has a trivial or non-trivial charge. In flat space QFT for example, gauge transformations that do not die off at infinity are physical symmetries and have important consequences like soft theorems and memory effects. $\endgroup$
    – Gold
    Commented Jan 7, 2023 at 2:40
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    $\begingroup$ For more details on this see physics.stackexchange.com/q/719053 $\endgroup$
    – Gold
    Commented Jan 7, 2023 at 2:40
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    $\begingroup$ Is non-triviality of charge the only indication for true physical symmetry? As a counter-example: in YM theory, I can have conserved current which is not gauge invariant, so not a physical observable $\endgroup$
    – paul230_x
    Commented Jan 7, 2023 at 8:31