I can only assume creating languages is part of the linguistics field, but is there a more specific name for the field, or the process?
It depends on exactly what circumstances you're talking about
Pidgin formation and creolisation are processes that occur when a new language is formed due to contact between two languages. The resulting language has features of both and often has a much reduced morphology than either. Pidgin formation tends to be short-term, as pidgins by definition lack native speakers, whilst creoles have them. In both cases, whether this constitutes "creation" rather than "evolution" is debatable
Another such debatable case is when a single language, or dialect continuum breaks apart into multiple distinct languages. This is extremely common historically, being why we have distinct languages in the first place. This is usually called breakup, but can also be called branching (in a tree model), or speciation (by analogy to biological evolution, and usually in the context of a dialect continuum)
If people set out to construct a language though, this is usually called conlanging (a conlang being a constructed language), although Tolkien referred to it (at least when pursued as an artistic aim in and of itself) as glossopoeia. Some conlangs even have native speakers (Esperanto has quite a few, and there was also a guy who tried to raise his kid as a native Klingon speaker, although as he was the only other Klingon-speaker the child new, they quickly rejected the language)
Conlangs are often split up by purpose into e.g. artlangs (made as an artistic pursuit), auxlangs (or auxiliary languages, intended to aid intercommunity communication), or englangs (engineered languages created in order to have some particular feature, such as Lojban's ability to be used in a syntactically unambiguous way). All of these categories can be used as verbs (e.g. someone might describe themselves as auxlanging if they are designing an auxlang)
In the context of academia you may also see toy, model, or nonce languages, which are usually very bare-bones, consisting only of exactly as many rules as needed to illustrate the concept at hand. E.g. the dual might be explained using a toy language featuring the words "teb" "apple", "tebu" "two apples", "tebya" "more than two apples". I don't know of any concise term for the creation of such models, and would probably just describe it as "making a toy language" or similar
Tolkien is said to have called it glossopoeia. Here is a useful survey of the word and its history.
Short answer: The discipline could be any.
A language could be made up so as to serve for a research experiment. I think that's what David Adger has done. https://davidadger.org/2020/02/16/invented-languages-and-the-science-of-the-mind/ He's a theoretical linguist, does syntax.
A language could just as well be made up so as to imitate/fantasize a particular natural language, for a purpose outside of science: Francis Nolan for example made the "Parseltongue" for Harry Potter movies. https://harrypotter.fandom.com/wiki/Francis_Nolan He's a theoretical and forensic -- and, in this case, commercial -- phonolgist.
These two are both artificial (made up).
As for natural language on the other hand, young children when they acquire language do "make up" their language, or, rather, make up "wrong" generalizations. Finally, another related phenomenon/concept to your question may be language change, which happens over time. There is a lot of research on both of these.