How to make the last network user username remain in the login field?
Ok! I have learned something. I will leave this here in hope it helps others who experience the same issue.

When setting up the AD binding there was an option about "create mobile account at login" which I left off. I assumed it was like a Windows roaming user.

However, it seems if you don't turn this on, OS X won't create a home directory for the user, or a local account for them.

When you turn it on, and login again, it does. Having the home directory/local account means you can turn the icon login option back on (so they don't need to enter the username), it means you can make them an admin of the machine, it means they can login when offline, and various other things.

So, turning this on has essentially solved a number of puzzling and frustrating things.