The names of dishes (and ingredients) are culinary trade jargon, just like the names of the techniques used to prepare them. In order to achieve unambiguous communication between chefs across temporal and cultural distance, terms like "cacio e pepe" have to be taken to mean one fixed thing—one platonic ideal dish that someone hypothetically made at some point, that all such real dishes are attempts to recreate. Otherwise chefs will just be talking past one-another.
Certainly, you can do things any way you want, and cooks can evolve dishes any way they want. Despite this, terms like "proper [dish]" and "authentic [dish]" still have a use.
A "proper" dish X is the dish that a professional chef expects when they hear X—the culinary-jargon interpretation of the term X. (So: "proper" nigiri sushi is "the nigiri you'd prepare if your practical exam in a culinary academy was to make nigiri.")
As stated above, this is a pretty objective concept; "proper" X will continue to refer to the same exact dish no matter what country you're in, or how long it's been since the dish was invented. (Tweak the dish? New dish, new name. We have a flat global culinary namespace. Yes, regardless of language, because it has to avoid collisions once translated into French. It's much the same as zoology's binomial nomenclature's flat global Latin namespace.)
An "authentic" dish X is the dish that a chef conversant with the culinary heritage that originated the dish X, would make, if you ask them for X. (This is not to be confused with authenticity of ingredients. A Ukranian immigrant making borscht with American beets is using inauthentic ingredients to make authentic borscht.)
(Amusingly enough, often an "authentic" dish X is the least "proper" X, because the dish has evolved in its cultural homeland since it was invented, and so the global culinary profession's concept of the dish has diverged from its homeland's evolving conception of it. The avant-garde and the authentic can overlap. But the avant-garde cannot, by definition, be "proper.")
...and then there's the term "traditional", which just means for a dish X that the chef who made the dish is a traditionalist—nostalgic for some bygone era (where the era they're nostalgic for is left unspecified, can change from chef to chef, and can change over time!) This term is, by contrast, pretty useless. Saying "give me a traditional X" is kind of like saying "give me a [git ref HEAD~1] X." Might work for you now, but won't work later. Chefs don't tend to use the word "traditional" much. (Instead, they'll speak of a culture having a particular culinary tradition, but will identify a specific period of that tradition if they want to pinpoint a particular dish they want to make.)