Hello @XSO,
Thank you for the very detailed information. I have checked your site and I could see that you are using classic sections. To have a better control over the background images, I would recommend that you use the v2 section element.
With v2 section just like in your test page, you can position the background image in anywhere you want. You can have top left, top center, top right, bottom left, bottom center and bottom right.

I also noticed that you are using an older version of X(5.1.1) and Cornerstone (2.0.6). Could you please update to the latest versions? Our latest versions are:
- X theme 6.5.6
- Cornerstone 3.5.5
This latest release contains fixes for several issues so be sure to check out the changelog (http://theme.co/changelog/).
_After doing the updates, always remember to clear all caches (if you are using WP Rocket, WP SuperCache or W3 Total Cache) when updating so that the code from the latest release is always in use. This will help you to avoid any potential errors.-
Please let us know how it goes.