I know cottage cheese as it's from warm temperature of milk and rennet, and ricotta cheese from high temperature of whey and acid.
So what is the cheese from boiling milk and acid, and does the leftover water from this and ricotta cheese have few meaningful ingredients? Has most of protein been extracted and could I discard it?
The cheese made from heating milk to 85C and then adding a coagulant is generally called Farmer Cheese, as it's the easiest useful cheese to make from whole milk. While this coagulant is generally an acid, it's still farmer cheese if you use rennet.
That's 85C, though. Neither ricotta nor farmer cheese get quite boiled. The only cheese where bringing the milk to a boil is common is paneer.
You will note that this is very similar to recipes to make cottage cheese. That's because the two cheeses are the same, up until it comes time to handle the curds. Farmer cheese is cut small, cooked, and drained, resulting in a creamy or chalky spreadable cheese, and cottage cheese is cut large, chilled and salted, resulting in larger, softer curds.
Note that a lot of the "ricotta" sold in supermarkets in the USA is actually farmer cheese, since it's made from whole milk rather than whey.
The whey leftover from either farmer cheese or cottage cheese is not useful for further cheesemaking. Due to the high-heat cooking, all useful solids have been removed from the milk. The whey can be used for other purposes, such as wet-packing feta and mozzarella or making bread.
Cottage cheese curd can be set with rennet, acid and rennet, or acid alone - the rennet coagulates more protein than the acid alone and gives a firmer texture. The reference chapter at the bottom explains further.
The whey after acid setting of milk will have some residual protein, though may be similar to the acid whey byproduct from yogurt manufacture where it's difficult to use for further extraction.
Acid-curd cheese process. From Guinee et al.
Fresh Acid-Curd Cheese Varieties.
T. P. Guinee, P. D. Pudja & N. Y. Farkye.
https://doi.org/10.1007/978-1-4615-2648-3_13
