Machete!
I've played with them all and machetes are always my favourite.
They are the Mora's of the big tool world imo. Something like a £12 tramontina latin or bolo will cut, chop and hack well above it's weight and cost comapred to any of the other options, while still being packable in size unlike the hooked slashers.
Axes and billhooks are good at their jobs but slashing weeds and thick brambles they will suffer, particularly an axe.
Another option people often don't consider is the Cold steel Bushman knife, with the socketed handle. It's a light weight uber tough knife that can be transformed into a hard working weed/bush whacker with the addition of a longer handle/haft! (bottom right of the pic, the black one is the bushman).
You are welcome to borrow my bushman if you like, give it a beating and try it out.
In summary, a cheap machete or about "18 will be ideal.. imo.