Hunter braids in the US have a LOT (i.e. 20+). Dressage usually less (12-15). I was always taught even for mares and odd for geldings as well (but then I always asked whether than included the forelock (LOL). And it depends upon the illusion you are going for as well.
Well, looking at Felix's mane even after the pull - it's going to take about 342 My understanding of the theory of button braids is you do as few or as many as you need to use up the mane in a way that looks pretty, and flatters the horse.
Proud owner of one Lunar acre! (Campanus Crater, The Moon)
Never been a hunter, so I've never counted. I'm also not a lazy braider and do not have horses where fat braids are flattering.
I just looked at some photos. I get 20-25 braids on my horses, and always get compliments from people watching the classes. It's different and flatters my horses necks. Go by that. Not by a set number or to fit in. Do what makes you happy and what makes your horse's neck look gorgeous.
(BTW, I also still sew my braids. I like how much tidier they are and how long they stay the way I want. I only did rubber bands as a kid when learning how to braid.)
I believe the last time I braided my guy, who has a THIN mane, I got 9 braids. I didn't even bother braiding his forelock, because he's worked so hard to get it to be respectable, I couldn't bear to mess with it.