What you have there is an eye hook, at least in this part of the country. In other areas it's called a winch hook or a cable hook.
It slips into the spliced or swaged eye on the end of a cable slides through the eye and seats into the eye when the button rests against one side of the eye. On the plus side, they're handy. On the minus side they can and do fall out, especially from cables on deck winches when going down the road. There was a time when every line truck carried one so the winch cable on the A frame or boom could be wrapped around a pole instead of using a sling. OSHA changed that!
You can determine what size cable it fits by measuring the mouth of the hook and deducting 1/16" if the hook ain't been stretched.
