keonas said:
Snapback is very much the most prominent term nowadays though.
I thought I'd add my two cents to this topic with some personal observations. Having grown up in the United States, we would typically refer to the type of hat found in the following posts as a snapback (modern usage):
post #3308935
post #3298084
post #3273926 (Perfect example)
Note that the bill of the hat is flat (or mostly so), and therefore wider than your typical baseball cap. It is what you might expect the "cool kids" of today to wear. While technically all hats with a snap in the back ought to be referred to as "snapbacks", I think that changing the name to something like that would bring some confusion for the above stated reasons. I think that companies like Tillys, Zumiez, and Lids popularized the term "snapback" in the US to mean this style of cap.
Last words:
post #3307215 would likely be referred to as a baseball cap in common speech, and not a snapback, even though it may very well have a snap in the back.