Hi, welcome to the forum! Glad you joined!
Either one should stop a fox if it is installed correctly. Foxes aren't that big, they should not be able to tear the smaller 19 gauge. The 12 gauge should stand up to a big dog (think 100 pounds) the 19 gauge maybe not so well. I don't know how big your birds or prey are, if they can get through a 2" hole or not. Snakes, weasels, mice and rats can.
By installed correctly I mean different things. How well is the wire attached to the supports? Can something pull the staples or whatever you use out? Many people use fender washers and screws to firmly attach the wire. That's not bad for the 1/2" but the 2" maybe more of a challenge. 3/4" poultry staples won't hold that well, the 1-1/2" fencing staples usually work really well if the wood they are nailed into is strong enough to hold them. Personally I like to screw strips of wood over the wire with the screws going through holes in the wire mesh. Drill pilot holes so you don't crack the wood strip but if you clamp it down tightly the wire is not going anywhere. I use a minimum of 3/4" thick and 1-1/2" wide strip of wood but nothing wrong with heavier or larger wood.
The other problem is that a fox or other things can go under or over a fence. A lot of the time they don't even need to dig to go under, they can squeeze through a very small gap. The way I stop that is to use an apron. I put a strip of wire mesh 12" to 18" wide around the run and attach it to the bottom leaving no big gaps. I lay it flat and bury it about 2", just take the sod off, lay the apron, and replace the sod. The idea is that the fox or other critter goes up to the fence, starts digging, hits the wire, and does not know to back up.
Do an internet search on "fox climbing fence" and watch a few videos. They can be enlightening. Most of those show them jumping more than climbing but you'll get the idea. There are ways to protect against this. Putting a roof over your run can work, whether solid or wire mesh. Vertically extending your wire mesh fencing a couple of feet above the top of your run so it bends over if something tries to climb on it works really well. Or you can add a wire mesh "shelf" at a 45 degree angle or flatter at the top of the fence so anything climbing has to hang from the top if trying to climb it.
Another method is to electrify the fence. That can be very effective. Follow the manufacturer's instructions.
Some people don't do nearly this much and just don't seem to have many problems. While others do a lot and a critter still finds a weakness in the defenses. Predators are a challenge in keeping chickens.
Good luck and once again
