Drupal unable to send emails using the contact forms [a host issue]

Like many of us, I am also facing an issue with sending emails from Drupal website hosted on Linux Host. From my tests I found that -
1) I can send and receive emails by accessing webmail of the domain's email account.

2) But when sending email using the contact from on the website, Drupal says email is sent successfully but emails are not received. Nor are the emails visible in the "Sent" box in webmail. The same email was configured on the site as well as contact form settings.

The error thrown by Drupal is as shown above:

"Unable to send email. Contact the site administrator if the problem persists..

Incidentally, I also tested on a WordPress site and even emails are not getting sent from WordPress's Contact Form 7 plugin.

So I ruled out Drupal as the issue or WordPress for that matter. It seems to me that it could be the host that is not allowing the emails to be sent. So I contacted my service provider.

And, here is the reply from my hosting service provider:

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