Considering the update to your question, you seem to have a sort of hybrid of two methods. The "Traditional" French method to make a cheese sauce, or Mornay, is described in the steps below with some additional notes at the bottom of alternate methods. If you go through this process to the "Bechamel" step, you'll get the white sauce often used for lasagna.
An alternate method, which uses kneaded butter and flour as described in your question is called a beurre manié. The difference between the traditional method for this and your method is that you place the kneaded mixture in a pan and start heating it and gradually add milk while a traditional beurre manié would be added directly to hot liquid as a thickening agent. Once it has reached your desired thickness, you can then add cheese to make it into a cheese sauce.
If you are heating your kneaded butter/flour paste so that it's actually melting down and cooking a bit before beginning to add the milk, then it's more similar to the roux preparation, though a roux doesn't require kneading the two ingredients together before putting them in the pan, though I don't suppose it would hurt.
A roux is the first step in a traditional French cheese sauce. It's the process of cooking together equal parts of flour and fat and is used to thicken various sauces and gravies.
Roux is flour and fat cooked together and used to thicken sauces. The fat is butter in French cuisine, but may be lard or vegetable oil in other cuisines. The roux is used in three of the mother sauces of classical French cooking: béchamel sauce, velouté sauce, and espagnole sauce. Clarified butter, vegetable oils, bacon drippings or lard are commonly used fats. It is used as a thickener for gravy, other sauces, soups and stews. It is typically made from equal parts of flour and fat by weight.
Different sauces will need you to cook the flour-fat mixture to a certain point. White sauces will generally cook until pale tan/yellow while others will cook until slightly darker and the general rule is, the darker the roux, the less it thickens but the more flavor it offers.
The fat is heated in a pot or pan, melting it if necessary. Then the flour is added. The mixture is stirred until the flour is incorporated and then cooked until at least the point where a raw flour taste is no longer apparent and the desired colour has been reached. The final colour can range from nearly white to nearly black, depending on the length of time it is over the heat and its intended use. The end result is a thickening and flavoring agent.
When you add milk to your roux when it is in the whiter end of the color spectrum, you get a bechamel sauce. The amount of milk added will determine the end thickness of the sauce.
Béchamel is traditionally made by melting a quantity of butter, and adding an equal part of flour to make a roux, which is cooked under gentle heat while stirring with a whisk. As it is a white sauce, care must be taken not to brown the roux. Then heated milk is gradually whisked in, and the sauce is cooked until thickened and smooth. The proportion of roux and milk determines the thickness of the sauce, typically one to three tablespoons each of flour and butter per cup of milk.
One tablespoon each of butter and flour per cup of milk makes a thin, easily pourable sauce. Two tablespoons of each makes a medium thick sauce. Three tablespoons of each makes an extra thick sauce, such as used to fill croquettes or as a soufflé base. Salt and white pepper are added and it is customary in Italy to add a pinch of nutmeg.
It is common to find lasagna made with bechamel sauce - though it doesn't have to be - many people make it with ricotta or even cottage cheese instead of the sauce.
While you certainly could call the final cheese sauce a Mornay, the traditional Mornay is made with Gruyere cheese though it can be made with other cheeses, including cheddar - even your addition of "extra fat" could be considered similar to the traditional addition of an egg yolk, which has a high-fat content, to a Mornay - it could also more specifically be referred to by the English name "Cheddar sauce" which is commonly used for Macaroni and cheese and other applications.
Other Cheese Sauce Methods
That being said, there are tons of different ways to make cheese sauce that range from this version (and the beurre manié) all the way to simply melting a cheese that melts well into a sauce with no thickeners at all. This is commonly done with special cheese products like "Velveeta" in the US.
Other options, like this recipe for nacho sauce, use thickeners in the form of cornstarch and add evaporated milk to help prevent the cheese from separating into an unsightly mess of solids and grease. See their testing process here.