This one hits it on the nose. The pronunciations in the OP are No. 3, possibly originally influenced by southern Italian dialectical pronunciations. But they are essentially Italian-American pronunciations today, not actual Italian.
I’ve been living in Padova for a good while and I still can’t understand people when they speak dialect. Good for you, but I think it only sounds closer to “standard” Italian because you got used to it.
I guess I think of it as a “j” sound because in French, “je” sounds like what I think “zh” denotes. Whenever I hear the sound, in my head I see a j because of that, even though now that I think about it, we don’t really use the j" sound for that in English.