in "the magnetic potential is gauge invariant". It differs for different gauges, so is not gauge-invariant, but gauge-dependent. A choice of A _is_ a choice o

in "the magnetic potential is gauge invariant". It differs for different gauges, so is not gauge-invariant, but gauge-dependent. A choice of A _is_ a choice of gauge. The theory using it (i.e. E&M Lagrangian) is what is gauge-invariant. This is a hyper-correction; in practice physicists apply the term in places adjacent to where it should be all the time.