Since we are trying to look at the security lug nut only these pictures are terrible. I can't really tell what's going on there.
But since you lost the key, it's time to just destroy these lock nuts without destroying the finish on the wheel as best you can.
When I did this I got a 12pt socket, found the one that went fairly cleanly over the lug, and then went down a size, and hammered it onto the lug. It displaces the metal and grabs on, and its especially easy if it's an aluminum lug, but these look like steel.
If that doesn't work you could try cutting a thin groove through the top of the nut through to the stud itself using a dremel/grinder/file, and then, using an impact screwdriver with a flathead bit, make sure it's set to go left and then add counterclockwise pressure as you hammer it with a large hammer.
You could also get a torch, and turn it on low and slowly add heat to the center of the lug to make either method more likely to work. Torch set on low so as not to burn your wheel paint.
Good luck