When I was searching the name of the Moon's largest crater, many websites said that it was the South Pole-Aitken Basin. But in some definitions I've found, the word "crater" is used to describe smaller impacts on the Moon surface, and since the South Pole-Aitken Basin has a diameter of almost 2500 km, I'm not sure if you could say that the South Pole-Aitken Basin is the Moon's largest crater.
The French version of the Wikipedia article has a "nomenclature" section, absent from the English version. It says:
Lorsque l'énergie cinétique de l'impacteur est suffisante pour atteindre le manteau à travers la croûte et provoquer des épanchements magmatiques, on parle de bassin d'impact et non plus de cratère d'impact.
Which roughly translates to:
When the kinetic energy of the impacting body is sufficient to reach the mantle through the crust and trigger magmatic outpourings, the term impact basin is used instead of impact crater.
However, there is no reference to back this claim, so I would take this with a pinch of salt...
The International Astronomical Union (IAU) is the official naming body in astronomy, and they also set the terms used to describe planetary features. For example, you may have heard of "Mons" (Olympus Mons on Mars, for instance) – that term "Mons" means "mountain" and is an official term from the IAU.
The IAU does not have any official nomenclature nor term that is "basin." Therefore, the term "basin" is purely colloquial/informal so far as official nomenclature goes.
In contrast, a "crater" is, basically, a circular cavity carved into the surface. There are several types of craters, including volcanic calderas which are craters, pit craters (collapsed lava tubes), and impact craters. Impact craters are craters that are formed when a projectile hits with enough energy to excavate that cavity.
Impact craters range in size from micrometers ("zap pits") formed by what we might think of as space dust or micrometeorites, to thousands of kilometers across. So, the South Pole-Aitken feature is technically a crater. So is Herschel on Mimas (the "Death Star" moon). So is Caloris on Mercury, Hellas and Utopia on Mars, etc. Even though these are thousands of kilometers across (except Herschel, that's about 150 km across), so far as official nomenclature goes, they are all just "craters."
However, sometimes colloquial terms are used more often than official terms. "Basin" is one of those terms, and it tends to be used to describe craters that are several hundred kilometers across or larger. So, all those I mentioned in the previous paragraph (again except Herschel) tends to be colloquially referred to as a "basin."
So, the South Pole-Aitken "Basin" is the Moon's largest known impact crater. Nomenclature doesn't matter, it is what it is and was formed when an object several hundred kilometers across hit the moon very early in the solar system's history.
(Full context: I am a planetary scientist who has studied impact craters for ~15 years, including having built the largest global crater databases for the Moon and Mars. I actually asked this question a few years ago at the Planetary Crater Consortium meeting a few years ago and the above was explained to me. Ever since then, I always refer to impact craters as "craters" in my papers, even if they are colloquially referred to as "basins," because that's the formal, proper nomenclature.)