king vs king-pawn

I'm new here and I posted this under game analysis, but it belongs here.

In the game geert-seuntjes I met an interesting situation which resulted in a draw (although I intended to abandon first).

I had only my king and my opponent his king and a pawn.

When his pawn was on b3 and his king behind it, I realized I could make a draw of it. When his king was on the 4th line I needed to stay on the b1 and b2 field, when he moved to the third line, I had to face him. 

What is more interesting is that I think that if the king of my opponent had preceded his pawn, he would have won. He made a mistake on move 54.

Is my analysis correct?

geert 

Hi Geert, I annotated your endgame below. K+P vs K is a fundamental endgame and well worth mastering completely.  :)

Thanks for this elaborated reply

geert 

The only thing that I have to ask is that the message is weird

Thanks for telling me that

Why is it so hard to read

..