Oh, very different plant! Yes, yours would have trouble with cold weather. I think of them as a kind of tomato, so it didn't occur to me that someone might call them "currants," any more than I would call other tomatoes "cherries" or "grapes."Currant tomatoes? These days they’re marketed as “Everglades tomatoes.” They’re a wild tomato from south Florida. Growing up we called them “currants.”
I was talking about the woody bush things that grow well in cold climates, and make red berries (redcurrants) or sometimes other colors (whitecurrants, blackcurents). It looks like they're in genus Ribes.