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.
Mathematical logic means many things, depending on context. In particular, it includes two related areas of study:
Using mathematical to study "logical" topics such as proofs, models, computability, and sets.
Using those logical topics to study mathematics.
The quote from Tao only scratches the surface of mathematical logic. It is analogous to calling the elementary facts about sets at the beginning of a textbook "set theory", when the true study of set theory goes much deeper.
The reason that logic is divided into "proof theory, model theory, computability theory, and set theory" is historical, which is to say it is not based on any kind of rigorous argument. Those four areas were developed during a similar time period, and were all originally motivated by certain foundational issues in math. Some of these areas and their subareas are still closely connected to foundations of math, while others are less closely connected.
Over time, as "logic" became its own subfield of mathematics, these topics became more and more fixed as "logic". The Handbook of Mathematical Logic in 1977 crystallized this division into four areas.
Another common feature of the four areas is a focus on formal languages and formal definability. This is not common in other areas of mathematics, where only natural language is typically used.
The four parts, though, are not intended to be exhaustive. Parts of category theory are closely related in spirit to the other areas of logic, even though category theory is not one of the four areas. Some areas of model theory are much more "mathematical" than other areas of mathematical logic. So the entire idea of dividing "logic" into four parts has to be taken with a grain of salt.