There are many recipes for alfredo sauce, using ingredients from pesto and soy milk to low-fat milk and cream cheese. What is the gold standard for alfredo? What properties indicate a traditional
Italian-style alfredo sauce? What type of pasta is it traditionally paired with?
According to Cooking The Roman Way by David Downie, Fettucine Alfredo is a traditional Roman recipe called "pasta del cornuti" (cuckold's pasta). What either Alfredo Di Lelio III, or Mario Mozzetti, depending on whom you believe, invented in 1914, was the dramatic tableside preparation of Fettuccine Alfredo, which is what made the dish a hit with visiting Americans in the 20's and 30's.
The tableside preparation is really what makes Fettuccine Alfredo what it is: the hot pasta is tossed with the butter and cheese in front of the diner, and then served to them immediately.
Again, according to Downie, the only ingredients of Fettuccine Alfredo are egg fettuccine, lots of butter, lots of Parmegiano-Reggiano, and (if necessary) a little salt.
Recipes which add cream or milk are Americanized recipes designed to allow restaurants to hold orders of Alfredo for a long time under heat lamps (and turn it into a gooey pasty mess). Italians, from my experience visiting Italy, rarely put cream or milk on pasta (a real Italian could speak up here).
Downie has a fun 3-page digression about the ongoing lawsuits between the Roman families who claim to own the name. It's worth a read.
Alan Davidson, predictably, says nothing about Fettuccine Alfredo. The Glorious Pasta of Italy likewise does not cover the dish.
Wikipedia supports Downie's story, except only attributing Di Lelio, and adding the tidbit that Di Lelio apparently called it "Fettuccine al burro" (fettuccine with butter), and the Alfredo name was appended later when it was copied in the USA. Wikipedia also says butter and cheese only, on fettuccine pasta.
So, to answer your question and the questions asked in the comments:
Now that we've established it's a modern American dish, I, an American, will tell you that authentic alfredo is cream, butter, parmesan, black pepper, and fettuccine.
It is made by melting butter in a pan, adding heavy cream and bringing to a simmer, adding cooked and drained fettuccine, tossing in grated parmesan and a few generous pinches of freshly ground black pepper, and serving immediately it's reduced to your desired saucy thickness.
Anyone who adds garlic or cream cheese is going to food purgatory! ;-) Or just enjoying a lovely variation of a creamy white sauce on noodles.
In my training as a chef, I have learned that there is no Alfredo with cream in Italy. The closest approximation is just butter and cheese with some added pasta water to thin it out.