If I select specific text and mark it as a header e.g. "Heading 1 " all the text will be wrapped in h1. Pressing "enter" doesn't work, only shift+enter creates line breaks. (probably that's the issue)
Marking field as "Allow multiple paragraphs" solves the issue.
I am glad that you figured it out. That's true that you have to enter shift+enter creates line breaks. You also must check "Allow multiple paragraphs" to add content in the Rich Text field.
Let us know if you have any questions.