But it's not taught that way.
It's never taught that way in US schools. Ever. It's always taught as an abstraction without ever tying any of it to real life. Ever. (repetition for emphasis)
It is taught that way if you have a good teacher. All my math teachers were excellent so we got lots of practical examples. But just like any skill, there is a lot of what one of my math teachers called "crank and grind" that you have to go through to internalize the skill enough that you can then focus on applying it.
With formal systems you usually have one definition that written to make it obviously correct and another that is more "pragmatic". In the case of propositional logic the obviously correct formalization is truth tables which are completely untractable to work with for large numbers of variables but are very simple conceptually. The "pragmatic" formalization is the logical connectives like and, or, implication etc that we normally think of as propositional logic. When they prove that propositional logic is sound they mean that all propositions give the same result as a truth table when evaluated and when they prove it is complete they mean that all truth tables give the same result as a proposition and thus propositional logic and truth tables both formalize the same concept.
There is no concept of "correct" in formal systems because it is inherently an informal concept meaning it does what it is supposed to do.