I think what kind of wire you use for your run depends a lot upon your situation. While most here on BYC advocate hardware cloth (1/4" x 1/4" weave), I don't think that is necessary in all cases and might even be overkill, expensive and only serving to allow nervous chicken keepers peace of mind. I see nothing wrong with 1" x 1", 1" x 2", or even 2:" x 4" welded wire. Unless you live in a hot climate where your birds might stay out all night, most of us lock our birds up at night behind fortress walls with wire sunk into the ground or surrounded by wire aprons with hardware cloth over windows and vents. Unless you have a problem with little varmints that might come around at any time of day, like rats or snakes, you don't need a run encased in hardware cloth. And, I have yet to see a raccoon, possum or other mid-sized critter come around during the day (since they are nocturnal) and reach through a welded wire opening just to snag a chicken by the neck and pull it through piece by piece. If you have a problem with bears (and yes, I have a friend who had a bear attack his chickens), it doesn't really matter what type of wire you use, does it?
One of my pet peeves on BYC (and knock me off my soap box if you are so inclined) is that many new members read certain :"gospel facts" and then repeat them so often trying to be helpful to others, that they get perpetuated. Even some old timers perpetuate the hardware cloth myth in every case because they have always done it this way. What may be best in one situation, may not always be what is needed in every situation. If money is no problem or you need to have the piece of mind, then by all means, use hardware cloth. If you have a budget or want to be realistic, take a long look at you local situation and then decide.