If you search in the MS store app for Age of Empires (Definitive Edition), what do you find?
The version that you find there now should be the latest client. If possible post back a screenshot from the search result in MS store. Also click … next to profile image and check for stuck or paused downloads and also check ‘My library’ and ‘Ready to install’, it might be listed there.
Maybe it got installed already on your system without your knowledge, so I would search in the Cortana search glass too you can recognise a slight difference between the logos. In my case Cortana search doesn’t even show the old client anymore although it’s still installed.
But the tile in my start menu still works.
Otherwise there is no option left other than to remove the old client first and thereafter install it fresh again from the MS store or to wait for that reminder message. I didn’t get it for some time already too.