I think there are several reasons. First, if you do it with an int, you're probably using up 32bits per value. You'd need 1 but waste the other 31, they needlessly take up storage. (Edit: Though your bool will most likely take 8 bit anyways due to other things.) And then sometimes it's nice to be expressive. So no one needs to remember if 1 is True or 0, or if True is greater than False, whether 2 or -1 map to True or False. And you end up in situations where either 2 equals 1, or True isn't always equal True. Or you do weird things to handle values other than 0 and 1 in the operations. All of that can be avoided and the code gets more readable with specific types.

There's nothing wrong with using ints, though. You just have to make sure the maths stays consistent.

code is read more often that it is written. booleans are actually just integers under the hood but explicitly using booleans communicates your intent. you're saying to the next guy that "if this is something other than 1 or 0 something has gone wrong".

when it comes to C specifically, the bool type was a pretty late addition because as you say, an integer can do the job. but it's very good for organisation, and it also allows the compiler to make some assumptions about how the variable can be accessed. if you know something can only be in two states, you can shove a whole bunch of them into the same byte of memory.

Here's a little secret: All data types don't actually exist. When your code is compiled down to machine code your computer just operates on collections of bits. Types are a tool we use to make code more understandable to humans. int and unsigned int tell us information about how numerical values are interpreted. bool tells you that only two values are going to be used and that the value will somehow relate to true/false. char tells us that the value is probably some sort of text. But again, everything is just stored as bits under the hood.

You can go even further and define new types, even though they might still just be numbers under the hood - you could for instance define a new type, e.g. Age, which is still just an integer, but it tells you the reader more information about what it is and what it might be used for. You might define a new type just for error codes, so at a glance you can see that this function doesn't return an int, it returns an OsError, even if the errors are actually still just integer values.

C does however play somewhat loosely by these rules, and to try and 'make your life easier' will often 'coerce' types between eachother to make everything work. For instance, integer types can be coerced to booleans - 0 is false, everything else is true (even negative values). Later on you'll find that arrays can 'decay' to pointers, and other fun surprises. Personally, I think this implicit coercion between types is one of C's biggest mistakes. If types exist to help humans reason about code, what does it mean if the compiler needs to silently change types behind the scenes to make things work? It means the humans were sloppy and the compiler can only guess at what they actually wanted to do (and this is the best case scenario! What happens if the compiler coerces types in places humans didn't expect? Bugs!).

There exist a spectrum of different behaviours in this regard, some languages are what we call 'weakly typed' (like Javascript, or to a lesser extent C) and other languages which are 'strongly typed'. Weakly typed languages will try to implicitly convert types together to make things work: In Javascript if you try and add an integer to text "Hello!" + 52 the compiler will try to implicitly convert types until something makes sense, for instance in this case the compiler would say 'oh, they probably want to add the text "52" onto the end' and will produce "Hello!52". Sometimes this is handy, sometimes it introduces bugs. A strongly typed language will instead simply refuse to compile until you make the types line up.

Sometimes you need to store a value that is either true or false and refer to it later or maybe update it. Maybe you have some complicated logic for when a loop needs to end that you can't really represent in the while condition. You can make the while check your bool variable and then set it to false when you are ready to end. Perhaps you have a function that returns a bool value and you setup your logic so there is a default return value but you do some calculations and sometimes you want to return the other value. it lets you have a single return statement in the function which can be easier to read.

You use boolean variables in the same way as other variables. The example you share is indeed contrived. In that situation you need no variables at all. That example is merely showing the syntax.

If u accidentally us the wrong type of variable it outputs en error when compiling. That's one of the main pros of statically typed languages and why the most reliable scripts are written in them to my knollidge. I'm sure there a languages out there where u can us boolean as a int or float but non i know. Not that i would consider my self a programmer but i asked the myself the same questions when i started.