Haha, oh, you won't believe what I found out! I feel so relieved! And to think I was looking at the wrong place for the whole day!
I checked the log after I restarted bind, and found an interesting line:
Aug 25 02:51:09 drustvo-mmm named: zone drustvo-mmm.si/IN: loading master file drustvo.zone: permission denied
Then I used ls -la, and guess what I saw? All zones had read permissions set only for root. That's what you get if you work as root.
It's working now. Thanks Noway2!