I agree that the hospital lab probably did not know how to culture a marine organism or recognize one when they saw one.
People who swim in the ocean get sea water in their mouths all the time, no problem. Even though the mouth is a warm moist mucus membrany surface. Probably shouldn't drink seawater, though.
Most likely source of infection is contact with poison or bacteria that colonize in a nice juicy cut or abrasion. Contact being the operative word. I am guessing that if one wore gloves while handling the zo's and anemones and rocks for that matter, it would not matter if there was tank water inside the gloves. Contact with the toxins , bacteria, and sharp stuff would be greatly diminished just by having that layer of latex or whatever between you and the solid objects in your tank. Sure there is bacteria in the water, but at reduced concentrations compared to the surface of rock etc. Dose is everything.
Probably a good idea to get the kind of gloves without powder or aloe or anything else inside. I am guessing short ones getting wet all over would be easier to manage than long ones that leak.
Just my opinion. Don't bet your life on it. If I were the guy with the thumb, I would never let my hands get wet again.