Hi, I posted a message in another thread asking about this:
http://forums.windowscentral.com/win...ing-mails.html
Later I found your message and you have described exactly the same situation. I also have a company exchange account and a Microsoft account in Outlook.com/Live. And a primary private account in Gmail, which I mostly use. According to my testing the problem only exists with Gmail.
When I create a new message it always shows the email address in the sender field but during sending process it converts to the sender name when using Exchange/Outlook.com. But not with Gmail. So I am wondering if this is a Gmail related issue? My speculation: Could it be that the Outlook mail app is cross-checking with the service provider the sender name? And with Gmail this doesn't work so it uses only email address?
I have Lumia 830, same OS 10.0.14393.321 and Outlook Mail 17.7369.40828.0. So the Outlook update since your message has not solved the problem. In the other thread it was mentioned that this began with the Anniversary Update. I am not sure but it is possible, I noticed it maybe a month ago.
It would be very nice if somebody could find a solution for this...