Your understanding is correct. Put simply, a contradiction is a sentence that is always false. More precisely,
A statement is a contradiction iff it is false in all interpretations.
In propositional logic, interpretations are valuation functions which assign propositional variables a truth value, so a contradiction comes down to having 0's as the final column in all rows (= valuations) of the truth table.
In predicate logic, interpretations are structures consisting of a domain of discourse and an interpretation function defining a mapping from symbols to objects, functions and relations on it, so a contradiction is a statement which evaluates to false no matter the choice of objects and interpretation of the non-logical symbols.
Take the expression $\exists x (x < 0)$, for instance: This sentence is false in the structure of the natural numbers, but true when we evaluate it in the integers, or under some none-standard interpretation of the natural numbers where e.g. the symbol $<$ ist taken to mean "greater than". The statement is not valid (= true in all structures), but it is not contradictory (= false in all structures), either: While it may be coincidentally false in some particular structure/the situation we're currently interested in, it is logically possible for it to become true.
On the other hand, $\exists x (x < 0) \land \neg \exists x (x < 0)$ is true in neither of the above three structures structures; in fact, it fails to be true in any structure whatsoever: No matter which domain of objects we take and which interpretation we assign to the symbols $<$ and $0$, the form of the statement $A \land \neg A$ makes it inherently impossible to ever become true.
To pick up your example, "The sky is red" is only coincidentally false in the actual world because our earthly sky just so happens to be blue, but it is possible to imagine a universe in which the atmosphere is constituted differently and the sky is indeed red: The sentence false in the real world, but it is not contradictory. In symbols, the sentence can be formalized as $p$, and will have a truth table with both a true and a falsy column.
On the other hand, $x \in S \land x \not \in S$ is another statement of the form $A \land \neg A$, and thus a contradiction: It is false in all structures, and thus also in our real-world conception of sets in standard ZF set theory. Its truth table has only 0's, no matter which value the component statements take.
The symbol $\bot$ is used to refer to a contradiction. And indeed, any contradictory statement is logically equivalent to (and can be transformed into, using rules of inference) both $A \land \neg A$ and $\bot$: All contradictory statements have the same truth table with only 0's in the last column.
Best Answer
If we want to derive a contradiction (if any) from $P ∧ (¬Q∨¬R)$, we have to consider the two cases forming the equivalent disjunction:
See Proof by cases: if we show that $(P∧¬Q)$ implies a contradiction and that $(P∧¬R)$ implies a contradiction, it is done.