This behavior is not unique to your domain trust configuration, it's this way with traditional domain deployments as well. Typically when you see the logon prompt it will show you the domain you are trying to logon to. If this is already the correct domain you don't have to prefix (or use UPN). If this is not the correct domain then you must specify it as part of the logon process.
If the users use a domain account to logon on to their windows machines you can add the URL to their intranet zone (or any zone if you configure it to allow automatic logon). This does not remove the domain requirement but will pass it silently in the background so the users doesn't need to see the logon prompt.
The other method is to enable forms based authentication. This will allow you to control the authentication inputs - however you can't use the automatic logon functionality as stated above.
For non-windows based machines like MAC you can configure password helper like KeyChain to store and forward the credentials. The password vault in windows would work in a similar fashion.