The extent to which it is "distinguishable" is subjective, but the difference is certainly there. It doesn't cost any more to mount a tire like this correctly (with the word "OUTSIDE" on the outside), so why not do it? If it didn't matter, the word "OUTSIDE" wouldn't have been placed there.
I will concede the difference is there. And I will agree that it is just as easy to mount it with the outside out. However it is not "dangerous" if it is not, like some here are suggesting. As the above poster mentioned, no tire manufacturer would knowingly produce a tire that could be made dangerous by simply having a tire jockey inadvertently flip it. Would it cause a slight, all but undetectable difference in performance in everyday driving? Perhaps, but doubtful.
Remember, we're talking streets and highway's here with speed limits. Not racetracks with cars running on the ragged edge. And if anyone hydroplanes ANY vehicle in the rain, on ANY tire, they are driving too fast for the conditions, period. But there is no "danger" involved if these type of tires are flipped, the way the second shop told the OP. They were unnecessarily over dramatizing the situation. The difference would be barely detectable, not "dangerous".