I'd happily use both ketchup and tomato sauce to describe the vinegary spicy stuff in bottles. There's certainly never a problem with it getting confused with other types of tomato(-y) sauce, because it's usually perfectly clear from the context. (I don't really talk about pizza in this much detail, but I'm fairly sure the stuff you get on pizzas is tomato purée anyway).

In terms of pasta, you can have tomato sauce on that (it's, you know, tomato-y) and then there's a bunch of different tomato-based sauces that you call by their names. Americans don't think marinara has fish in it?? THIS EXPLAINS SO MUCH.