Those look like ¾” poultry staples. They work pretty well to tack the wire in place but I would not trust them over time. With wood expanding and shrinking they can eventually work loose. A strong raccoon or big dog might be able to pull them out. That’s a neat job, by the way.
I personally prefer the uncreased. I’m an engineer and am going to go technical on you. Sorry in advance. When something pulls or presses against the wire the creased is going put a tensile force on the staple, it will try to pull it out. With the uncreased it will put a shear force on the staple. That means it pulls from the side not from above. You’ll get a stronger connection with the uncreased.
To make that a whole lot stronger, make it prettier, and to cover the sharp ends, I suggest you cut strips of wood maybe ½” to a better ¾” thick to cover those ends. (Called furring strips) Drill pilot holes to keep the wood from splitting and screw those over the ends of the wire. Make sure those screws go through the holes in the hardware cloth. Clamp that down pretty tight and that wire is not going anywhere, creased or uncreased. If you use thinner wood that is really dry it can sometimes split but if you put a fender washer on the screw it will spread the pressure and really stop the splitting.
You did not leave yourself a lot of room for this with the uncreased but looking at your workmanship I think you can manage to get the screws through the holes in the hardware cloth.
Good luck!