Fae - also spelled fay* - is from the 12th century old French, likely from the older Latin Fata, meaning spirits of fate, and Williams suggests it entered French as a term for Celtic goddesses later shifting to women of supernatural power, then to an adjective meaning roughly enchanting, and finally to the place of Fairyland itself (Williams, 1991). This initial use for the place of Fairy is how the term enters English in the 13th century and we see it developing as both an adjective describing things with the nature of that place as well as a term for beings from that place. Briggs suggest that the initial adjective form may have been fay-erie, to indicate something that was enchanting or had an enchanting nature (Briggs, 1976).
In the oldest English sources we see fairy used as an adjective as well as a noun and this adjectival use continued for hundreds of years. For example, in Milton's 17th century work Paradise Lost, book one line 781 he refers to 'fairy elves' where fairy here is an adjective describing the elves. The noun is something of a catch all for any being from the land of Fairy or with a fairy-like nature and we see it used synonymously with elf, goblin, and incubi. This 16th century example from a poem by Alexander Montgomerie illustrates this well with fairy, elf, and incubi all being used interchangeably: "The King of Fairy, and his Court, with the Elf Queen,with many elvish Incubi was riding that night.". The terms don't have a fixed description or meaning beyond 'of Fairyland' and an implication of the enchanting and supernatural. Williams perhaps summarizes this variety of application best: "...fairy in particular, but more generally any supernatural name, is necessarily amorphous, and...from its earliest use in English...no single meaning has ever been paramount." (Williams, 1991, p 457). The term is used less as an adjective now but still retains it's use as a noun, indicating and Otherworldly being; it has also had secondary pejorative meanings over the centuries of both a promiscuous young woman and a homosexual man.
The meaning of the terms, applied to Otherworldly beings, remains vague through today with applications as an adjective and noun for both a place and beings from the place, although the application as a noun is the main one. We can find examples of fairy with both of these usages across folklore, modern anecdotes, and academia. Patricia Lysaght discusses the Bean Sidhe, an example where fairy is used as adjective**, in her book 'The Banshee'. The Fairy Investigation Society's 2017 Fairy Census offers examples of fairy as applied to various described anecdotal accounts. In some demographics the word fairy has become hyper-specialized to indicate only a type of small winged sprite, however across many other demographics the word retains its older broader meanings. This dichotomy of use by different groups means that context may be required in order to understand what the word means within any source. An academic paper using the word fairy is likely to be adhering to the broader meaning, as are occurences within folklore or traditional belief, but personal use or use within a specific group may follow the specialized meaning. This is an important distinction as the meanings have drifted so far from each other as to be nearly antithetical in nature now.
Fairy has multiple spellings across the written record because English had a non-standard orthography until relatively recently. This means that words were spelled in any way which might phonetically convey the sounds of the spoken word. Hence we see fairy as everything from feirie to phary to faerye. There are 93 different variant spellings noted by Williams with fairy being the most common at 724 occurrences followed by faery at 131, fayry at 55, and faerie at 49 (Williams, 1991, p 459). In current academic and folklore usage fairy is the usual preferred spelling, however as with the specialized meaning of fairy gaining popularity in some niches there has been an effort by some people to distinguish fairy from faery with the prior supposedly indicating twee, Victorian fairies and the latter supposedly indicating real or legitimate fairies. Similarly there has also been a push in some demographics to use fae as a term to indicate Otherworldly beings generally where fairy is used to mean only a specific type. These spelling and semantic issues, as touched on in the previous paragraph, can cause confusion in communicating between people or groups ascribing different meanings to the terms.
It should also be noted that fairy and fae in modern usage are English language terms and have only existed as such for about 700 years. These do not reflect Christianization as Western Europe was Christian for several hundred years prior to fae coming into French (arguably with a strong pagan connotation initially) but rather the evolution of the languages, particularly English. There were and are non-English terms within the cultures that now use fairy in an English language context, and these terms pre-date the word fairy but often have related or parallel meanings in context. As previously touched on the words elf and fairy are used interchangeably and that likely stems from the Anglo-Saxon term aelf which predates fairy but describes a similar type of being who was also equated later broadly to fairy, goblin, and incubus (Harper, 2020). In the same way in the Irish we see the Daoine Sidhe or Aos Sidhe [people of the fairy hills] or sióga whose name intrinsically implies that connection to the sidhe, the fairy hills or Otherworld. The word sidhe - modern Irish sí - like the word fairy indicates both the place (fairy mounds) and as an adjective things with the nature of the place hence sidhe, fairy hills, but also slua sidhe, fairy host, or cú sidhe, fairy hound, and in modern slang sidhe can also be used to refer to the beings of that place. Every culture will have its own terms like this, for which the English fairy is simply the best equivalent term.
It should also be noted that in many places there is a prohibition about using the term fairy and euphemisms are used instead. Euphemisms go back to at least the 14th century and can be found in across Celtic language speaking countries, as well as in older English material. One 16th century example from England uses the term Fair Folk in Latin, pulchrum populum (Green, 2016). The term Good Neighbours, in Scots, can be traced back to the 15th century. The concept behind the use of these terms rests in the belief that calling them fairies offended them and so one would want to use a term that was appealing or positive in case fairies passing by invisibly overheard the comment.
This summarizes the pertinent information relating to these words, and hopefully may offer some clarity to the subject, which is admittedly opaque.
* fae and fay not to be confused with fey, a Norse originating word for someone or something doomed or fated to die
**Lysaght's book is primarily focused on Irish language terms for the Bean Sidhe, however she does touch on translations of these terms which reflect the use of fairy as an adjective
Harper, D., (2020) Fairy Retrieved from https://www.etymonline.com/word/fairy
(2020) Elf Retrieved from https://www.etymonline.com/search?q=elf
Williams, N., (1991) The Semantics of the Word Fairy: Making Meaning Out of Thin Air; 'The Good People'
Briggs, K., (1976) A Dictionary of Fairies
Green, R., (2016) Elf Queens and Holy Friars