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
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