crowmanyclouds is correct that the second one could be called a Kaiser blade, or a bill or a bank blade or a gazillion other names. And the first one could be called abrush thinner, and many, many other names for that as well. And there many other variations in construction.
But the generic name for all these types of implement is just “brush hook”.
As for what they are for: they are designed to clear light woody vegetation. So they get used in weed control, for making temporary roads, constructing firebreaks. Pretty much anywhere that you want to remove plants that are too big to mow and too small and numerous for an axe or chainsaw.
Not to disagree with any of the above (there will be any number of valid names for implements like this). In the UK, those would be called a long handled billhooks.
A briar axe is another variation on the brush hook. It’s similar to your fist photo, but the blade is longer and usually narrower with less of a hook. It’s lighter and easier to cut a wide swathe but less useful against larger saplings.
I’ve seen the second sold in hardware stores as a “bank blade”. It’s used for brush clearing, particularly along irrigation ditches and such. Hence the “bank”.