The 'fairie' spelling is used intentionally to distinguish from the fairy tales in the modern sense and to include creatures grimmer than those pixies commonly implied nowadays by the 'fairy' variant.

I think this term could be used for tales involving water sprites, wood sprites and other nature-inspired spirit presences.

Given the number of distinctions that need to be made, it may be best to keep the Russian term.
The following links give some idea of how the term bylichki is used in relation to other terms:

Among the latter are the bylichki (sing. bylichka), tales mainly about the lesser demigods and spirit-beings, wood demons, water nymphs, spirits of the dead, who populated the familiar universe of the Russian peasantry. This form of memorate, or tale about events that had supposedly taken place in real life and were 'remembered' by the story-teller, shows deeply entrenched patterns of belief about the relationship between the natural and supernatural world in the traditional rural community. Bylichki are still being recorded today. The folk tales known as skazki (sing. skacka), on the other hand, are pure fiction and lack a sacral dimension. Yet here too, especially in the 'wonder' or 'magical' tale (volshebnaya skazka), there are mythical layers encrypted in poetic language. It is in the wonder tales that the frightening and enigmatic Baba-Yaga appears. The byliny (sing. bylina), which mostly relate the exploits of the early heroic defenders of Kievan Rus, blend myth with history, while legendy (religious legends) and dukhovnye stikhi (sacred verses), in which figures from the Old and New Testaments, saints and hermits meet with ordinary folk, blend myth with Christian piety. The last three forms, in contrast to the bylichki, ceased to be part of a living tradition about one hundred years ago. Echoes of myth may be heard in many other folkloric forms. Especially important are spells and incantations, in which Russia is particularly rich.
http://www.utexas.edu/utpress/excerpts/exwarrup.html

Bylichki are short and told in the first person or about someone known to the speaker. In vivid and emotive language, the tellers underline the veracity of the narrative by describing a landscape, people and objects familiar to the listeners (this is particularly evident in G--'s narration).
It has become common in folklore scholarship in Russia to determine the level of belief in supernatural phenomena by the narrative type through which information is conveyed, the scale running from simple statement of an actual belief to bylichka (narrative relating a personal experience), to byval'shchina (fictionalised narrative with vestigial relationship to actual beliefs) to skazka (understood to be purely fictional). The same supernatural figures may occur in all of these, though each type of tale reveals its own characteristic themes, motifs and structures. In the opinion of one of Russia's leading folk tale scholars, by the second half of the twentieth century, "the beliefs themselves have disappeared irreversibly from folk perceptions" and "bylichki and byval'shchiny have either disappeared or are undergoing transformation, evolving from memorates ... into skazki ... Only here and there do they continue to exist in the memory of elderly people" (Pomerantseva 1975, 5-6).
http://findarticles.com/p/articles/mi_m2386/is_1_111/ai_6268...

Short stories, supposedly true ones http://www.thefreedictionary.com/parable but I would not exclude a fib as any true story, as usual, is modified when being told from mouth to mouth during a long period of time.