Thank you for your reply. I have tried to rewrite .page-header,.item-title, h1 and h2 headers and now in fact the right google font (latin-extended) was rendered. This is great news.
Thank you!
It seems that the correct google font is applied only to the new menu items and new articles. The old ones render the wrong one.
Now, do you know why the correct font is not applied to the existing .page-headers and .item-titles?
It would be nice if there would be some possibility how to set all existing data so that it use the right font. It would solve a lot of work.