I have Facebook set up through my school email account and for the longest time, the notifications would appear in the Facebook app and also still in my email - WHEN I logged into my inbox on a computer. Never at any point did they show up in the inbox I have on my BB for my school email.
My guess is that this is simply how the app operates - it intercepts the notification when it reaches your live.com inbox on your phone and redirects it to the Facebook app. If you do not want the emails left over in your inbox when you log into it on a computer, you can go into the Facebook app > Options > Uncheck "Leave email notification messages on mail server".
Hope this helped