@HansH:
It’s strange that your main navigation is displaying with every word capitalized. I recall you sending a reference image for the ‘Navigation’ settings yesterday. In the image, you have three options for letter case: ‘uppercase,’ ‘lowercase,’ and ‘capitalize.’ To make the main menu appear in lowercase, please select the ‘lowercase’ option, as per your site’s requirements.
I’ve attached the reference image below for your convenience:

This will adjust your main menu to lowercase. If you’d like your submenu to follow the same style, please add the following CSS.
body .wp-block-navigation .has-child .wp-block-navigation__submenu-container .wp-block-navigation-item .wp-block-navigation-item__content {
text-transform: inherit;
}
Additionally, the two-stage changes (logged in/logged out) are likely occurring because many plugins disable caching when you’re signed into the WordPress login. To resolve this, please try clearing the cache and check if the issue persists.
Kind Regards,
Sujapati