Problems with Gmail in default Postfix configuration

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Recently I moved some of the sites to a new hosting. The previous server had Ubuntu 12.04 with Exim4 and I forgot how to configure them.

On a new server I installed Postfix with Ansible:

    - name: Set up Postfix to relay mail
      debconf: name=postfix
               question='{{ item.question }}'
               value='{{ item.value }}'
               vtype='{{ item.vtype }}'
      with_items:
        - { question: 'postfix/mailname', value: '{{ ansible_fqdn }}', vtype: 'string' }
        - { question: 'postfix/main_mailer_type', value: 'Internet Site', vtype: 'string' }

After that I started to receive the next delivery reports from Gmail: