Display domain URL in browser of WP installation

Hello @all

My problem is very similar to the problem of user “bamme” in this post: https://www.ads-software.com/support/topic/using-htaccess-to-point-domain-to-subfolder-how-to-install-wp?replies=4

However, my problem is a bit more sneaky:
I installed my WP site in a sub folder of my root directory of my web space. It can be reached via https://www.firstdomain.com/subfolder. To make it more comfortable for my users to visit my site, I bought a second domain and instructed the DNS provider to point this domain to the WP installation (so in short: https://www.seconddomain.com points to https://www.firstdomain.com/subfolder). This works quite fine.
Now, if someone visits my site via seconddomain.com, my WP site is displayed, however the URL in the browser says https://www.firstdomain.com/subfolder. Is it possible to change the settings of WP, thus only seconddomain.com is shown in the browser’s URL field?

I’ve found this ressource on www.ads-software.com > Using_a_pre-existing_subdirectory_install. But in this guidance does not consider the second domain.

Can someone help me?
Thanks in advance