Interestingly, my definitions of various tomato-based products isn't terribly connected to the purpose any more (but then, this is from the person who routinely says "blue green-bag", and suffers no dissonance whatsoever).

So, Pasta Sauce can go in pretty much anything, and is pretty much purely related to jar sauce. I have, on occasion, used Pasta sauce to make tomato-based rice meals, which are extremely delicious, and I highly recommend. If it's going on Pizza, pretty much without exception it's tomato paste, generally from one of the teeny tiny little cans.

Tomato Sauce and Ketchup are entirely synonymous. There's no distinction in my head whatsoever. But Pasta Sauce and Tomato Sauce are completely different items - you couldn't possibly mix the two terms in my head.