Yury, the leashes in your photos only work with axes designed specifically for those leashes. You can’t buy one of those leashes, slap it onto some other brand of axe, and have it work. I have an old Stubai like the one in your last photo. There is a peg screwed into the wood handle that limits how far the ring can slide. Replacing the leash requires removing and replacing that peg.
The right way to attach a leash is through the hole in the head of the axe. You can use a buckle, or you can just use a water knot tied from webbing. This is much more reliable than the silly peg screwed onto the handle, and it lets you use any leash on any axe, as long as the leash is the right length for the handle. I use the Black Diamond leashes on my technical tools, and I just tie a leash out of webbing for general mountaineering.
The Black Diamond Slider and Lockdown leashes allow you to adjust the wrist loop so you can’t accidentally drop your tool when you need it most. The Black Diamond Slider
has a metal tube that slides up and down the leash to adjust it. That’s why they call it the Slider.