According to this list, "Jos." for girls/women could have also possibly meant 'Josephte," as in Marie-Josephte Corriveau.
It's interesting to note that, whether right or wrong, the cited list uses "Jn-Bts." for "Jean-Baptiste," but your "J. Baptiste" is also most likely used here for "Jean-Baptiste," as mentioned by Vincent in his good answer.
Used as it is here with "Augustin," (usually a male name), "Amb.Augustin", as mentioned by Cl-r, most probably stands for "Ambroise(-)Augustin", as in Saint Ambroise-Augustin Chevreux.
"Amb." by itself could stand for "Ambre," as mentioned also by Vincent, but being as short of a name as it is, "Ambre" probably would not need to be abbreviated.
(Tongue-in-cheek observation with absolutely no merit and made with absolutely no offense intended to anyone: Since you mention that it's unclear if "Amb.Augustin" is for a male or a female, maybe the "Amb." is short for "Ambigu(e)" or even "Ambivalent(e)" as in "Ambiguously [Gay] Augustin(e)" or "Ambivalent(e) Augustin(e)!)
(I will gladly delete this final observation if even just one person finds it offensive and requests me to delete it or flags it as offensive.)