Just my two cents worth....
It seems to me that you should learn the Bb fingerings.
If, after you've been playing for a while, someone comes along and asks you to play the Eb Cornet in a brass band and hands you an Eb, you'll always be playing in F cornet unless you "transpose". That is, of course, unless they've got a spare Db Cornet laying around.
If someone were to hand you a D trumpet to play in an ensemble you'd always be playing in E trumpet.
Anytime you see a written B (third line) you'll want to play it with first and second valve which is not correct for a non-transposed part (the majority of what you'll be playing).
But that's only my opinion.