Original Alfredo is pretty much an American invention. There is a restaurant in Rome that makes "pasta al burro e parmigiano" but that's pretty much it. Americans took the dish, put cream and shit in it and gave it that name. They can keep it imo.
In Italy pasta Alfredo is more of a meme than anything else, and "pasta al burro" is made pretty much only when you are sick.
So you just make pasta with bechamel and parmigiano and drink caffè macchiato. You're fine, really.
Aside from the fact that, imo, there are better sauces for your pasta, you are not doing anything egregious, plus you can call Alfredo whatever you want. Nobody knows what it means anyway.
PS: coffee made with a moka pot in Italy is just called coffee. Espresso is made with an espresso machine that works at high pressure.