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.