How to set up a reverse proxy server for changing the link (not only the domain) to different services?

One thing that would help greatly here is to include your sanitized NGINX config(s). This will help with context and give a better picture of how NGINX will process incoming requests.

The nginx.conf can be found in /etc/nginx in most cases.

I am also including a blog post, I think may help, if I am understanding your use case correctly.

Creating NGINX Rewrite Rules