Odd that asking for additional server information. Per the screen grabs, your iOS device is setup using Exchange. When I setup Exchange on my Mac, only issue I had was that I needed to enter a app-specific password generated by the Outlook website (2-factor is on). Never asked for anything else other than login name. But, could be due to I never had a Hotmail or Live account, started with an outlook.com address (clean slate).
Then again, wouldn't be surprised in that there always seems to be a disconnect with Mail on Apple OSes. From persona experience, iOS Mail does Google IMAP well, where Mac Mail is a mess with regard to Gmail.
A possible option is to move over to outlook.com for your mail and create a mail alias back to the old Live account. Might not even have to do the alias, since maybe MS will handle that behind the scenes.