In the quoted passage you posted, they are talking about email clients and client networks.
The link you provided points to “Hacked website or CMS.” And yes, a normal web server—or really any kind of server other than a mail server—should not communicate directly over port 25. Instead, it should use port 465 or 587 with authentication to send email notifications.
An email server, however, does need to communicate over port 25 in order to exchange emails with other mail servers.
Port 25 → used specifically for email server-to-email server communication (this is how mail transfer agents deliver mail between mail servers).
Ports 587 and 465 → used for email client-to-email server communication (this is how your mail app submits outgoing mail to a mail server, with authentication).