How to update the theme in the right manner?

As David mentioned above - the reason the layout looks different is several resources are not being loaded. This is because they're still trying to reach the CDN: https://www.screencast.com/t/fynXkJzAHvUW

You can see this by right-clicking and choosing "Inspect" - then look in the "Console" tab (the screenshot above).

If those resources (CSS files) are returning 404 errors, your website will look different as it's missing a bunch of CSS.

This is something your hosting should be able to fix on the staging site - it's not something we are able to fix, as it has nothing to do with the theme - it has to do with how the CDN/hosting is set up.

It likely wouldn't be an issue on your live site, as the CDN is set up and working correctly on it. Of course, you'd want to perform a full backup before updating just in case.