Ustadh, Ahsan Allahu Ilaykum, when the scholars mention that a wet dream is one of the signs of puberty, is what is intended specifically, having a wet dream (with semen gushing out), or is what is intended that semen gushes out no matter if its from a wet dream, or other than that

Ustadh AbdulAziz al-Haqqan

Yes, when semen comes out, then he has hit puberty. Keep in mind that pubic hair and armpit hair are signs of puberty as well.