EWay transaction receipt not sent when member signs up using front-end form but works in the CiviCRM back-end

Using the EWay payment gateway if a new member signs up using the front-end online form then they receive a CiviCRM transaction receipt but not the EWay transaction receipt that is supposed to come through as well. If a credit card membership is added to an existing contact in CiviCRM's back-end (and "Send Confirmation and Receipt?" is checked) then both the CiviCRM and EWay transaction receipts are emailed to the new member.

On checking with EWay they report that in the former, problematic case, a billing email address is not being received by them, although a billing email address is clearly there in the contact record.

So there seems to be a difference in what billing data is passed to the EWay payment gateway API between the front-end and back-end of CiviCRM. On a side note, the CiviCRM email transaction receipts are also worded and formatted differently and contain different information, between back and front end.