• Apparently, Joséphine had a spike in popularity. Probably because
  of Joséphine Bonaparte , I’m not sure. I made my own family tree and I
  could not find a lot of Joséphine in them but I don’t really see
  other possibilities. Joseph was also a popular name for men.
• J. Baptiste : Yes, Jean is the most likely but it’s not possible to
  be 100% sure.
• Amb: It could also be Ambre, but it is not very common. Otherwise, in most case, as Cl-r mentioned, Amb stands for Ambroise.

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.)

On a complementary note, in Québec in the last five years you have about 10-20 baby girls per year named Joséphine. Generally speaking, considering the historical influence of the catholic religion over that society, you can expect the Bible and a practical book such as a missel to have been relevant in any past era. As liturgical books adapted for an audience, missels may vary in content (or order thereof) depending on origin. But of particular interest is a more or less uniform summary calendar which links days of the month to the name of saints; for instance the 7th of December to Saint Ambroise, Évêque (details).¹ This pool of names should provide insight, as it sure did in the past…

_{1. See Canada’s catholic calendar specifics(compare to France, contrast with Orthodox; Anglican). Different missels could also be further compared. See for instance this, and latin versions (1962, 2002). Unrelated, but could be useful considering genealogy, see also contemporary statistics and name of places.}