Learn English – What do you call people who like word puzzles

While several terms loosely fit the criteria (such as a culinarian), a culinophile or reagenteur speaks more to my tastes (if you don't mind portmanteaux).

As for what one might call the group, taxonomy or classification are appropriate when ascribing the term to a family derived from a particular set of criteria. Plenty of additional terms are available in online Thesauri, if you're curious.

Consider the colloquialism, tire monkey.

monkey: (slang) a menial employee who does a repetitive job, as in code monkey, grease monkey, phone monkey, powder monkey. Wiktionary

I would assume that the term can be perceived as offensive and/or derogatory, depending on context.