Just to add a "post script" to the above, not only will your actual sending address be what is shown in the recipient's message index list of his inbox, but also when the message is viewed in a preview pane or opened and viewed in its own dedicated window. Furthermore, even though the reply-to address should only come into play and populate the recipient's reply message's "To" field when the recipient hits the reply button, there is no assurance or mechanism to prevent the recipient from manually changing the "To" field" back to your original sending address.
Again, a post of the raw headers from the message as received by your recipient, and perhaps even from the message in your sent folder (Mail not set up on this particular computer so can't see if "Reply-To" is in sent headers, seems like it should be, but this computer only has Outlook Mail configured which apparently lacks this functionality, so can't check this right now), in order to determine whether the "Reply-To" header is being removed somewhere along the way, would be helpful to further troubleshoot your issue.