Okay, I think I fixed this issue. Just so I don't have similar issues like this in the future, when there is an update, am I supposed to download the "rest_files" and upload all of the plugins, etc. via ftp?
Plugins do not change often, this is why it can be separated from rest of extensions and packed in rest_files. If there is a problem with it like for example now after completely new joomla version than I'm trying to inform when this update is necessary.