Using New Outlook on Windows for personal email, calendar, and contact management
Again, there is no difference between a Hotmail account and an Outlook.com account. Any account using an address ending in @hotmail.com is an Outlook.com account because Microsoft renamed the Hotmail service to Outlook.com back in 2013. This is also true of accounts whose addresses end in @live.com and @msn.com as well. So anyone whose address ends in any of those domains log into Outlook.com to receive their mail, either using a web browser or an email client like classic or new Outlook, Thunderbird, eM Client and others.
I cannot think of a scenario that would produce what you describe unless you're using POP accounts on the devices that show more email. Could you verify the account type on the old computer and the phone. What client do you use on the phone? Is the phone Android or iOS?