The article did a really poor job of explaining the logic.
> It's strange to me that there are 5-6 options for each exchange, instead of a canonical name.
The point isn't to assign them canonical names, but to define a mapping from the two letters at the start of a word to two-digit codes. These letter-pairs are chosen so that meaningful words can be formed with them (that's why there is no QX, because no English word starts with QX) and that they are distinguishable (which is why there is a BA but not BE, BO, or BU: it would be hard to distinguish BAldwin from BEldwing, BOldwin or BUldwin).
There isn't a single canonical name because the purpose of the code is to allow cities to define their own area codes with memorable names that vary based on the location. One city might use UNiversity (86), MUseum (68) and CEnter (23).
But using UNiversity or MUseum wouldn't make sense in a different city that doesn't have a local university or a particularly notable museum. But they can still use the same area codes with the mnemonics TOwn Square instead of UNiversity (86), OVerlook instead of MUseum (68), or CEdar Avenue instead of CEnter (23).