Baking – Ingredients or techniques to a more solid lasagna

The general process of making lasagna is:

• Cook the other ingredients - brown the meat in a pan (maybe with seasonings); if you're using fresh tomatoes, cook them; if you're using canned tomatoes, drain the extra liquid out. The important thing is that everything is cooked, and there's not too much liquid left, since that'd make the lasagna messy at the end. This does mean it's faster with canned tomatoes, since you only have to drain them, not cook them.
• Cook the noodles as you would any pasta. Be sure to stir, since big flat noodles stick together more easily than most. (There are also no-boil lasagna noodles, but you've already bought yours, and I doubt that's what you have.)
• Layer the ingredients in the pan, starting with sauce on the bottom (otherwise the noodles will stick to the pan), then noodles, then alternate layers of the other ingredients and noodles and sauce until you run out or your lasagna dish is full.
• Bake it until it's heated through (the ingredients are all already cooked).
• Let it cool for a bit before serving; this helps keep it from falling apart when you cut into it.

When you go looking for recipes, though there are tons and tons of them, they'll pretty much all look generally like this. They'll just have variations in ingredient proportions, seasonings, order of layers, and so on. You shouldn't have trouble finding one that matches up with the ingredients you have, if you want a recipe to follow carefully. As long as it generally matches the steps I listed, it should be fine.

Lasagne is the Italian name for the noodles used in a lasagna casserole. So it would be technically incorrect to use it for a casserole made with a different type of noodles. And I'm not aware of any other use of lasagne noodles, so while you will probably have to call your soup "lasagne soup" if it contains them, and Italians will also use the term "lasagne al forno" when they want to be specific about the dish. But in practice, I have never seen recipes for anything else made from these noodles, and Italians shorten the dish name to just "lasagne" too.

Else, any casserole made with lasagne noodles should be called lasagne. The other ingredients don't matter much - there are versions without bechamel, without meat, etc.

Pasticcio is a wider category of casseroles which doesn't have a strict definition. It is the Italian spelling, you will also see other variants such as pastitsio. It usually has noodles, but not always.

Lasagne is a subtype of pasticcio, and the most common pasticcio in Italy - note that on the Italian Wikipedia, "pasticcio(cucina)" redirects to "Lasagne al forno".