Sauce – Making Tomato Sauce from Tomato Paste

In the US at least, common canned tomato products include:

• Paste, cooked down tomatoes, to the point where they are scoopable with a spoon but will not flow. Very thick, like peanut butter. Often sold in six or twelve ounce cans.
• Pureee - cooked tomatoes that have been--well--pureeed, but are mostly at their natural density; also called crushed tomatoes.
• Sauce - cooked down, strained tomatoes, a little thicker than tomato soup. May include herbs, spices, or some flavoring in addition to pure tomato product.
• Diced - solid chopped tomatoes, usually in tomato juice.
• Whole - whole cooked (usually peeled) tomatoes, usually packed in tomato juice. These will still have the seeds.

I wouldn't worry too much about your results not matching the picture.* If you got enjoyable results, enjoy them. However, some notes can be made about your concerns.

The bacon certainly isn't going to disappear -- no matter what you do (short of blending), you will always be able to see those bits.

The onions in the sauce shouldn't really be caramelized, but they should be thoroughly sweated -- without coloring -- so that they are soft and just starting to be sweet. If you'd like the onions to disintegrate more, you can cut them differently: either dice them finely or slice them latitudinally. There are fibers in an onion which run from top to bottom, so if you slice from pole to pole, you get the whole length of it. If you cut across the fiber, the slice will break down much more when cooked.

the water from the tomatoes evaporated without leaving anything of substance.

It sounds from that like you may actually have cooked it down too long. You can cook tomatoes all the way down to paste, if you like, but in this case you want to stop before that happens, leaving a fair bit of the tomatoes' juice, enough to coat the pasta when you mix them. It sounds like the sauce did thicken, and then went past that, to drying out.

Keep the time that the recipe gives for the tomatoes to cook down in mind only as a very loose guideline. Much more important is the results that you see on your own stove. The person who wrote down that recipe has a different pan, different burner, different kitchen... Of course, to know exactly what is right (for both your equipment and your preference), takes repitition, and thus the time-based guideline to put you in the ballpark.

I'd suggest just trying it again, paying particular attention to the consistency of the sauce after you add the tomatoes. Taste and evaluate it frequently as it cooks, and take it off the heat when it looks and feels the way you expect.

As an aside, if you're interested, the official recipe for the sauce is available on the website of Amatrice (the town), although it's in Italian, of course.

*For one thing, the presentation is amateurish -- the cooked pasta should be mixed with the sauce, rather than just being topped with it. (And it's entirely possible that the picture is not even the product of this recipe.)