It comes from my earstwhile piano teacher. She has (I suppose she still does) a Peters album with Solvejg's Song and this one, which is a transcription by another hand (I mention it in the tags): Walter Niemann, of whom I know nothing. I have a copy, which I hang on to for dear life, as it does not seem to be available elsewhere.
Ah, Niemann, a well-known name. he transcribed a lot of stuff, mostly for Peters I think.
Do you mean the piano sounds bad because it sounds bad or because it is out of tune?
I don't think it is out of tune. It's just the sound of the high register in the first few bars that made me cringe a bit. After that, the sound is quite passable. But I sure hope you'll get access to a better instrument.
One version of Solvejg's Song is to be found in Grieg's transcription of Peer Gynt suite No. 2, while the other is in his second series of Piano Pieces After His Own Songs op 52. I had chucked the copy I had that was coupled with the Cradle Song after I had obtained op 52. What was my horror when I realised that it was something else! What was my relief when I realised the one I knew was there, all the time, in the Peer Gynt album and I had never bothered to look!
Checked my books and you're right, there are two versions, which in fact I've played both without realizing they were different
The difference is very small though, the op.52 version just a little more fleshed out.