Not born county means just that.
I think the 1861 census has been mistranscribed as this appears to be the same family.
1851 Census HO107/2402/39/13
Living at Brunswick Terrace, Gateshead
BECKETT Thomas Head Married 36 years Railway Agent Born Durham, Durham
BECKETT Mary Ann Dau 9 years Born Durham, Durham
BECKETT Isabell Dau 4 years Born Gateshead, Durham
COLLINS Elizabeth Sister in law Unmarried 34 Born Durham, Durham
COLLINS Robert Nephew 10 Born Durham, Durham
COLLINS Walter Nephew 9 Born Sunderland Durham
DIMLEBY David Visitor Unmarried 32 years Railway Clerk Born York, Beverley
Marriages Dec 1840
BECKETT Thomas Newcastle 25/358
COLLINS Isabell Newcastle 25/358
Cando
Edit to add marriage