It's a technically a misnomer. However, everyone knows what it means.
Double action originally meant the weapon could be fired two different ways--cocking the hammer or pulling the trigger. When most people say "double action" they mean "trigger cocking" (i.e. pulling the trigger cocks the hammer). The pull is longer, stiffer and, at first, more awkward until you get it sorted out (you get to shoot--yeah).
Mossberg put a longer, heavier trigger on their shotguns to sell to mall ninjas whose policicrats are worried about liability from cops shooting each other or the wrong persons with NDs. Rather than give their men training, policicrats always seek magic swords over instruction. You see, training costs money whereas magic does not cost as much.
If it sells for Mossberg, Remington, Winchester, HK, FN, etc. will market one as well.
However, if you follow Rule #3, which as a TFLer you must now do always, you do not need the policicrat trigger.
If you like it, buy 4 of them and get to school. See you out there.