I was going to say - Lakeland to me as a Brit is a store, lake district sounds way more proper.
Lakeland makes it sound like a theme park!
That took me a minute - it's pronounced 'lake-lund' (as in French lundi), which doesn't sound so theme-parky!
For us the shorthand term was always 'The Lakes'. I never really heard anyone use the term Lakeland, though there is a South Lakeland District Council so maybe some people do.
I've definitely heard people refer to Lakeland the place.
But it's unusual, mostly people seem to just refer to it as "the lakes".