My only concern with just using staples is whether or not they are strong enough to hold the hardware cloth into the wood if something tries to push or pull against it. Do you have racoons? They will pull and work on something until it comes loose.
We have hardward cloth on our coop as well, although we don't have coons - just river otter and weasels. If you have to overlap the hardware cloth (like we are having to do on the front of the coop) think about using heavy duty zip ties or wire to keep the hardware cloth together.
I used staples similar to what you pictured, only about 1 1/8 " long, and spaced them about every 2 - 3". Since I'm very concerned about coons (we have quite a few on our property) I'm also going to screw large washers over the hardware cloth about every 12". My husband thinks I'm crazy... but I'll be danged if I'll let those coons get into my coop/run.
We used the heavy poultry staples for most of it but for places that were likely to be attacked we used the screw/washer/rubber gasket things they make to screw down corrugated roofing. I had some, they were really handy. I'd advocate a screw and washer of some kind in the heavy stress areas, corners, the base, etc.
And nowhere that there was going to be predator pressure did we only use hardwire fabric. It always went over 1x4 welded wire. So far, so good. The fact that the outside pen is electric is probaby why we've had no problems at all though. I still feel safer knowing I made it like Fort Knox.