Hi @Pbalazs89,
Actually, @christian is correct, and one thing I noticed is in your video, there is no language flag option that allows you to translate/duplicate a header. This means, what you’re creating is one header for all language in which its styling is not loaded, since it’s basically assigned to the page ID. And there are two pages of home page, one for reach language, hence two IDs as well.
Hence, I checked and duplicated the header for Hungarian, then tried to switch to CZECH through preview language flag, and it says there is no translation. And the copy-existing is empty as well which shouldn’t be, since the existing header for Hungarian exists.
It’s empty due to javascript error caused by grecaptcha, as shown here

So I went ahead and deactivated the plugins and the existing translation now displays it okay
Again, this is not related to the issue. The issue I’m seeing why it’s not displaying it the same as the preview (actual page versus builder preview) is cache. I saw a similar case and not limited to header, and when it’s moved to another host, it displays the front okay the same as on the preview.
But one thing for sure, it works okay when the header is set as global instead of assigning it per page. Could you try checking it again? I set X Header that I created as global (please note the plugins are still disabled).
Thanks!