Propositional logic is about Boolean inferences like "If A then B, A, therefore B". The link you provided expands this to the more complex predicate logic, of which syllogisms (discussed in the article) are special cases. E.g. "Everything that is X is also Y. This is X. Therefore, this is Y."
Though I would also note that most introductions to symbolic logic do ommit the (in this context) most important part: On how to translate natural language arguments to formal logic, and the other way round. This task is very much non-trivial. A few textbooks aimed at philosophers do this, though most don't.
Actually, propositional logic is a lot more than that. The core is really about proof theory, natural deduction, and axiomatic systems that form the precursor of first-order logic and the foundations of model theory.
Well, natural deduction is one specific form of proof theory (it's a kind of proof system), but there are others. Moreover, proof theory and model theory are independent (neither relies on the other) and various logicians would say that we don't really need one of them, though they would disagree on which. Proof theorists tend to be skeptical of the necessity of model theory, model theorists tend to be skeptical of the necessity of proof theory. Mathematicians tend to be skeptical of both.
But yes, to the degree that any such theoretical parts are part of predicate logic, they are already part of propositional logic. The former is an extension of the latter. Perhaps similar to how higher-order logic (simple type theory) extends or generalizes both propositional and first-order predicate logic.
By "natural deduction", I assume both of you refer to the system which is related to "sequent calculus" ? [1].