There is no name from Bartok. It is simply shown as “Quasi adagio” in the tempo marking. I don’t know where the popular name “A Winter Tale” came from, though it is common. This is simply the 3rd song in his book called “For Children, Volume I”. The names are very confusing.
This is yet another miniature by Bartok out of his collection of pieces named “For Children”.
“For Children” (Hungarian: Gyermekeknek) is a set of short piano pieces[composed by Bela Bartok. The project was originally begun in 1908 and completed in 1909, and there were four volumes.
Each piece is based on a folk tune, Hungarian in the first two volumes (42 pieces), and Slovak in the last two (43 pieces).
In 1945, Bartok revised the set, removing six pieces which used tunes which had been inaccurately transcribed or found not to be original folk tunes, and substantially changing the harmonization of a number of others; the collection, now of 79 pieces, was reissued in two volumes (of 40 and 39 pieces). These pieces were written for students to play, and they progress in difficulty through each half of the collection.
However, in spite of Bartok’s genius, his ideas about what relative beginners can play and express were horribly flawed.
The moment you hear such music played by very fine pianists you know two things. First, the music has to possess something special that attracts these great players. Second, when great players are attracted to music there will also be something very difficult about the music. If the music does not have a lot of notes and does not obviously demand some kind of advanced technique, there will be even more musical demands.
And that means that in the hands of students such music is always going to sound woefully inadequate. The bottom line is that any music that demands sensitivity and expressiveness is difficult. It’s just not obvious to listeners. Geniuses like Bartók with very idealistic ideas of what young children or inexperienced pianists can play are very poor at writing music that actually works for students in their first few years.