Thanks to everyone for the responses.
Hungarian: Neked is koszonet Peter! Remelem nem banod, hogy tegeztelek, de az interneten ugysem szamit a kor.
Yesterday I played the entire piece fluently with the exact interpreation I was thinking in my head... it was like magic! I was all alone and I could really feel the mood of the piece just like someone suggested, and I was 100% focused. It really helped me that I was all alone, no distractions, no one to play for and to be nervous about.
I played for 3 and a half hours, began learning new pieces, and still there wasn't a single time I zoned out. This experience makes me think that the best way to practice for me is alone. I guess, I am not confident enough to play when people are around. I make mistakes, I accidently acent notes too much or not enough, but yesterday I hardly made a single error in 3 hours and I played my pieces up to speed.
I'm guessing, there are big challenges for concert pianists away from the piano too, such as controlling their anxiety, since this obviously leads to errors. Performing for a crowd seems like a different form of playing, than just playing for yourself. Anyway, thanks for the advice, I think I am on the right road now as far as my concentration is concerned.