Those are completely different from the first ones you pictured. Those use old-style "cleats" (from before the days of clipless pedals) which just have a slot that fits over the rear cage plate. When you pull the straps tight they're very secure, but you have to reach down and loosen the strap to get out (that's why clipless was such a great advance).
I don't know if anybody still makes that type of cleat, or if it works on current shoes -- maybe some some track-specific stuff. In the really old days you nailed the cleat to the wooden shoe sole.
FTF: Bata Bikers? LOL . I think I still have a pair sitting in the back of my closet.
DB: You needn't ditch them, depending on your use. If you remove the tab thing at the front (and re-attach the clips), they'd make perfectly good pedals for street shoes or sneakers, if that's what you need.