How can you be certain that your notes are correct? This position is a bit complicated to work out at a glance, but as I see it there is only one possible winning plan for black here, consisting of the following:
play a well-timed ...d4, and after White plays Kxd4 Black will play ...Kc6 and force White's king back.
However, White has pawns that they can use to manipulate tempi, and in order to actually break through in the center with the plan I outlined above Black actually needs to force zugzwang TWICE! This is because even if White has to move their king from d4 after ...Kc6, White can still move their king to e3 and counter Black's ...Kd5 with Kd3.
And so, the question becomes rather simple: can Black engineer this position so that they have TWO more spare tempi than White instead of one? We note immediately that at most one spare tempo can come from the queenside pawns with ...a7-a6 (if White is careless and plays, say, a2-a3 prematurely), meaning that Black needs at least one spare tempo from the kingside as well.
If White is to move, then White will just pass the move over to Black with 1.Kc3, Ke3 or Kd4. This does not really affect the evaluation of the underlying position, since the only real winning plan is still a successful ...d4 push. Thus, we may consider ways for black to force the issue.
Black has, in total, four pawn moves and a bunch of king moves. The king moves clearly don't lead to any substantial progress, so let us consider the available pawn moves.
1...a6 and 1...h5 are immediately off the table, since they waste Black's precious opportunity to create an extra tempo, and thus we may consider the other pawn moves as the only possibly winning moves.
If Black plays 1...g5, then White can respond with 2.h4 with the threat of playing 3.h5! This would force Black into going for the draw, since otherwise White could easily create a passed pawn. Thus, we get that after 1...g5 2.h4 Black needs to play either 3...g4 or 3...gxf4. Out of these, 3...gxf4 is the only move that can actually work. But even this move fails, since after 3...gxf4 4.gxf4 we see that Black has only managed to create ONE spare tempo; after 4...d4 5.Kxd4 h5 6.a3! a6 7.Ke3, the position is indeed drawn. From this, I make the conclusion that the position is drawn after 1...g5.
Left to consider is 1...a5!? This move looks quite odd at first glance, but it's actually not completely harmless. If White plays 2.bxa5?, then black responds with 2...b4 followed by ...Kb5 and ...Kxa5. This would indeed be winning, so White needs to play 2.a3 here (2.Kc3 axb4+ 3.Kxb4 d4! wins for black). Thus, we may conclude that the line has to go 1...a5 2.a3, and here we run into some issues as Black. Now the pawn on b5 can never be protected by a pawn on a6, so Black has to be careful about allowing Kc5. Moreover, as long as Black's pawn on a5 remains undefended Black cannot play the ...d4 push and get the desired zugzwang with ...Kc6, since this would let White just take on a5 and win. Thus, we can conclude that Black NEEDS to play either ...axb4 or ...a4 before the ...d4 push. But this means that Black has zero spare tempi on the queenside. Because of this, I conclude that the position is drawn after 1...a5.
To summarize: your notes are definitely wrong, and the position is a draw no matter who makes the first move.
Edit: As indicated by other answers, there is one serious attempt involving a king march to provoke h4, followed by engineering an ...a5 push to gain the extra tempo needed in order to make the ...d4 break work. I did not consider this option when I posted my answer, and although my conclusions are still correct (the position is still a draw after some precise moves from White, see Carlo Wood's answer to get an idea of the lines that need to be considered), this answer is definitely incomplete. I guess it just goes to show how tricky pawn endgames really can be. Take care, and remember: don't make any hasty assumptions about pawn endgames!