I recently purchased a Barit handlebar bag, which is from Ortlieb's "economy" company Racktime. This uses the Ortlieb mounting system and the bag is very similar in construction to their more expensive waterproof bags but is not waterproof. Anyhow, my handlebar bag snaps looked exactly like your pictures when I received it. I separated the snap, and pushed the snap part that had pulled out back into the hole where it came from. That's the best I can describe it. I have not had any problems since then but I have to say that most of the time my top is not snapped closed because it fits snuggly enough that even when off road riding it has yet to come open on it's own. I don't know whether any of this makes sense but hopefully it will help.
On a side note-
1) I would have preferred a velcro closure too and may put one on there myself at some point.
2) The way the snaps attach to the bag is just a friction fit through the hole in the plastic. If you continue having problems I would put a little super glue or epoxy into the hole before sticking the snap