I was going to say - Lakeland to me as a Brit is a store, lake district sounds way more proper.
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.
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!
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".