You are right that cows give you whole milk which can then be passed through a centrifugal separator to separate the milk based on density to get all the different grades. Thickest is different types of cream and the thinnest is low fat or no fat milk. The milk is then homogenized to make the fat and clumps of protein to mix fully without separating. The term skimmed milk is from when they would be able to skim the layer of cream collecting on top of milk when it sits which is not possible with the homogenized milk that is common today.
Butter is even higher fat content then you get with a separator. To make butter you start off with unhomogenized cream. This have small clumps of fat floating in it. By churning it, for example with a kitchen mixer, you get these clumps to join together into one large clump. You can then take this large clump of butter out and are left with buttermilk. The process is very similar to how you make whipped cream although in that case you stop just as the fat joins together leaving pockets of buttermilk and air. If you continue to work the whipped cream it can turn into butter.
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