we are effectively talking about some peasant dish here - some eggs from the smallholding, a bit of cured meat, small grate of cheese (if you are lucky) and of course some pasta. There is no one right set of ingredients.
Yes. And no.
Yes, because it's a dish made from the generic ingredients you list.
No because, but if you go back through old books on Italian cuisine, there is a dish called gricia (mentioned upthread) with its origins in the 17th Century or earlier, as something cooked by seasonally migrant/itinerant shepherds (who might even have been Swiss, rather than Italian!) who carried pecorino, cured pork (referenced as guanciale rather than pancetta), and lard, with them on their travels. Given the greater meat content of pancetta (and presumably, therefore, cost) compared to guanciale, it makes sense that it would be lard and guanciale they carried. Thus, if you're chasing the original, the cured meat should be guanciale, the cheese pecorino.
If you go through Artusi (generally accepted as a good reference of Italian dishes up to the 19th century), there is no reference to carbonara, and no references to egg being added to cured meat in a pasta dish. The follow-on to gricia came in the 18th Century with the invention of tomato sauce: sugo all'amatriciana, which is gricia with tomato sauce, but still no egg.
It is generally accepted that the first references to carbonara don't appear until the closing years of the Second World War or even the 1950s. There is a suspicion that the egg component came from US soldiers's rations (so processed egg, probably dried). In which case the specific ingredients, if you're chasing the original, do matter, and they would be guanciale and pecorino, and K ration processed eggs. Adding delicious, free range eggs, laid by lovingly tended chickens in cosy barns, is just a modern affectation.