What is mathematical logic?

Mathematical logic is a strange beast.

It is a perfectly ordinary branch of mathematics whose goal is ... to study mathematics itself.

Thus, the different branches of mathematical logic are devoted to the study of some basic building blocks of mathematical practice : language, model, proof, computation.

The names and scopes of areas of mathematics are not always crisply delineated. In this case set theory is a bit of a grey area. There's an argument for considering it part of the broader subject of mathematical logic, but there are many set theorists who wouldn't consider themselves logicians.

Likewise, whether recursion theory is part of logic or computer science depends on who you ask.

That being said, the two descriptions are not in conflict. The four subfields that Wikipedia lists are all ingredients of the study of "what a rigorous proof is" and what rigorous proofs can and can't achieve.

Proof theory and model theory are both unquestionably part of logic.

Set theory is part of the common language of mathematical proofs -- it is used as a general way to speak about the things of actual interest in whatever your field is. Figuring out the appropriate rules for how set theory can be used therefore (arguably!) belongs as part of the study of common features of mathematical proofs in general.

Recursion theory is the study of mechanical computation, and is -- in addition to being the foundation for computer science -- an important technical tool for proving famous results of proof theory, such as Gödel's incompleteness theorem.