Spätzle originates from a region spanning Austria, Switzerland and southern Germany. I can tell you that here in southern Germany, Spätzle is definitely considered a sort of pasta. You can buy premade, prepackaged dried Spätzle, and it is always on the pasta shelf. So I'd say it is correct to refer to it as pasta, at least for historical/traditional reasons, even if it doesn't meet some technical criteria for pasta (or does meet some for dumplings).
As for technical criteria, I don't know of any decision criteria for either category both are loosely defined in my head by enumeration of their elements. Maybe there are some official criteria, but kitchen taxonomy isn't as hard defined as e.g. biological taxonomy, and it varies by region/country. Consider for example, the German word
Braten, which is generally translated as a roast. But while a German cook may consider a piece of meat cooked in a dutch oven on a stovetop a kind of
Braten, any cook from the Balkans will tell you that this is not a roast, as it wasn't made in a "real" oven, but in what is, in his eyes, a pot. So would you define the meat as a roast or not? I'd say that here regional tradition is best. If it was a German recipe, you are free to call it a roast. If it was a Balkan recipe, you'll have to call it something else, depending on the exact recipe. Similarly, for Spätzle, I'd say just go with the tradition and call it a pasta.