That's pretty much the recipe that I've been using forever. I don't usually measure anything, but the spices and the ratio of sauce to paste is the same.
To me the difference between spaghetti sauce and pizza sauce is that spaghetti sauce is cooked and pizza sauce isn't (although the tomato paste is previously cooked). Spaghetti sauce starts out thin and is reduced by simmering.
My wife makes incredible spaghetti sauce. It simmers slowly for many hours. I tried it on pizza once and, while it wasn't terrible, it was way too rich and flavorful and overpowered the other flavors in the pizza. To me, the sauce is the primary flavor in pasta, but in pizza it's just one of the flavors that make up the whole flavor profile of the pizza.
For some pizzas, I just use crushed tomatoes with some spices. Mostly pies that I use fresh mozzarella on.