Glad you figured it out! I also faced some issues with it, during my initial mail server setup.
Just make sure you use the mail.specific-website.com as the host in your SMTP settings, instead of the central mail.server.com (Mail Role Server Hostname). This way it works more dynamically even if you ever move the Mail Role to another VPS with a different hostname in the future.
Also something else you might need, is to manually generate an SSL certificate for the mail subdomain of a website, provided that the DNS propagation has been completed.
You can do that by browsing to:
Website Name -> Security tab -> (click the 3 dots) -> Request Let's Encrypt certificate for mail.