Sauce – use of xanthan gum in tomato ketchup

Emulsifier will make two or more element blend together, for example the egg in the mayonnaise recipe will act as an emulsifier.

A thickener will simply make something thick, as you wrote, cornstarch is a thickener.

Personally, I would use neither cornstarch or xanthan gum in a chill sauce; seems to me that it is a shortcut instead of letting it cook down properly (but I might be wrong).

Fresh tomatoes are insanely watery, so you're starting at a pretty big disadvantage here. Trying to fix it with a thickening agent alone might not be the best plan.

That said, if you want a short answer: use tomato paste, whether homemade or storebought. It'll thicken and improve the flavor. Watery tomato sauce usually has watery flavor, not just watery texture, so any way you thicken besides tomato paste is likely to leave you with a thick sauce that still lacks flavor somewhat.

First off, try to make sure you're using more pasty less juicy tomatoes. It's hard to give very specific advice since this depends a lot on exactly which varieties you get. But everyday supermaket tomatoes are generally too juicy, with the plum tomatoes often (but not always) being somewhat better.

From there, the most common options leave something to be desired:

• Remove some of the juice before cooking. This costs you some flavor, though.
• Cook longer to reduce the sauce. This costs you the fresh tomato flavor.
• Puree the sauce, so that the pulp gets spread around and thickens the liquid. This doesn't fix everything, but it helps a bit. You get this for free with longer cooking and reasonably small chopping, but you can also use a food mill, or grate the tomatoes, or find another way to thoroughly smash them.

So the best option, if you're looking for a totally fresh tomato sauce might be something like:

• Make your own paste (nice and thick and rich). It's easiest to reduce without burning in the oven, but you can do it in a pot too with some care. This is the time-consuming part, but if you do it in bulk, maybe not so bad.
• Make your own quickly-cooked puree (so you keep the fresh flavor as much as possible). If you have a food mill you can save a lot of time here by just hacking up the tomatoes and bringing them to a boil as fast as possible and cooking for 5-10 minutes, then using the food mill to get the skins and seeds out.
• If you want chunkiness, either mix in some fresh tomatoes just before serving, or if you want the chunks cooked, accept that you'll need more paste to make up for their water.
• Add paste until it's thick enough.

Then since that's all a bit time-consuming, you can save time by buying paste (it's cheap and easy). If you don't have a food mill, peeling and seeding is kind of a pain if you're making larger quantities, so you might consider just chopping reasonably finely and tolerating the skin and seeds.

There's a much longer writeup, "How to Make the Best Tomato Sauce From Fresh Tomatoes", of essentially this technique with an awful lot more attention to detail at Serious Eats. I personally find the lazier versions to be a better balance of time spent and resulting sauce quality, but you might think it's worth it.