...and ketchup is a jam

Ketchup is my jam.

If we are to accept the madness of treating tomato as a fruit culinarily (although, your comment makes a strong argument for not doing so), would it be a jam, jelly, or…?

How many jams, jellies, marmalades, or confitures do you know of that contain vinegar, savory spices, and salt? In a recipe calling for jam, would you replace it with ketchup?

There are several chutneys that include vinegar. They aren't in the same category as you listed, but the parent included an ellipsis, which I'll include chutney under.

Some "fruit preserves" also use vinegars and other acids. Which leads to the Venn diagrams of the definitions of pickles, chutneys, and preserves (fruit) all have a massive overlap and ketchup is right there in the middle as pickled tomatoes, tomato chutney, or tomato preserve depending on how much you want to classify it is a fruit, vegetable, or "it doesn't matter". (I would at least argue that "preserve" is the more accurate fruit term over jam or jelly.)

>In a recipe calling for jam, would you replace it with ketchup?

if the recipe was for French fries and jam: yes.

Tomatoes comes from South America so couldn’t have been on the Pompeii pizza

I think they were saying that the argument "fruit doesn't belong on pizza, thus pineapples don't belong on pizza" is invalid because most people agree that tomatoes go on pizza, and tomatoes are biologically fruits.

You didn't read the article at all, didn't you? Yet here you are commenting like an expert.

The very first sentence: Whilst it looks like a pizza, this image from a Pompeian painting from 2000 years ago obviously can't be, since some of the most characteristic ingredients are missing, namely tomatoes and mozzarella.