Hi Danail,
If I use the editor in the way you've demonstrated in the video, it works for us too. That is, simply adding the button and saving the page. As I mentioned in my post, the issue shows up when you have added the tag and then add a web part that causes a post back when setting the properties of the web part. Here are the steps that I've followed to replicate the issue:
1. Create a web part page
2. Add a button tag to the editor content area (I have used a subclassed RadHtmlField as mentioned at the begiing of the post) of the page and save it. After you have saved the page with a button tag in the content, go back to editing the page
3. In edit mode, add a web part to any of the web part zones. Select the "Edit web part' option from the added web part and in the web part's editor, set any properties and click the OK button. This will cause the page to post back. Now inspect the content of the editor. This is where, in our environment, the issue occurred.
Thanks for looking into this,
Rajesh