Is canned “tomato sauce” already cooked? The can says the ingredients are tomato puree, garlic, etc.  I suspect that cooking turns puree into sauce.  But then why does it change again when simmered for half an hour with ground meat and chili seasonings?

The meat's already browned prior to adding the tomato sauce, so I was thinking that we wait until the tomato is cooked. But my wife says the tomato is already cooked.  So what is happening, and what do we call this when cooking?

To reiterate, suppose the components are not “raw”.  But you still have to simmer it all together for 30 minutes.  What do you call that, since you’re not moving from a state of being uncooked to being cooked?
Yes, in some sense, the contents of a can of tomato sauce are already cooked. But "cooked" is a pretty vague term, and we know that there are different degrees of it. Did you cook your onion on a low heat until it was soft, or until golden, or until caramelised? Did you cook your steak until it was rare, or medium, or well done?

Likewise with tomatoes, if you blend them up and then cook them for ten minutes they're not going to taste the same as when you've cooked them for three hours in a low oven with the lid off. Lots of chemical reactions are going on inside the food as it's cooking, and some of them only happen over rather long time periods.

So what you're doing when you're putting the "cooked" canned tomato sauce in with your "cooked" meat and leaving it for a while is... cooking. The combination of ingredients plus more heat and more time will cause further changes and different flavours to emerge.

Also, canned tomato sauce will only be cooked enough to preserve it in the can - it's not been cooked to the point where it's going to be at its best for eating. That part's up to you.