I'd like my tinymce to just apply tags so that my clients never screw up the look of a website. I notice when I am applying e.g. a Header tag, it adds the styles of my Header from the CSS file (although I am not sure if it loads from another, different CSS or...).
Is there a way that i disable that, I just need clean markup/tags?
TinyMCE creates clean markup. It just uses and loads the editor.css file from your mysite/css directory. By default, the editor.css also includes typography.css which is where your typography styles (for h1, etc.) are defined.
If you don't want this behavior, just remove the import statement in the editor.css file.
This doesn't create different markup though. It's just influencing the way content is being displayed in the CMS. Usually, it is considered a nice feature when the content is displayed the same way in the editor as it's being shown on the live website.