I believe this to be John & Elizabeth's son William. His children's names, make the 1785 marriage of his possible grandfather look more likely. Maybe some of the records for Williams siblings are missing?  
1851  Orton, Cumberland, all born Carlisle. No Marriage?
William Mckenzie 27 (1824) Pot Dealer 
Jemima wife 22, John son age 3
1871 
https://www.familysearch.org/ark:/61903/1:1:V51K-XXB?lang=enGRO registrations, mother MCKENZIE/MCKINZIE/MCKINZEY  
MCKINZIE, JOHN    
GRO Reference: 1848  M Quarter in CARLISLE  Volume 25  Page 72
MCKENZIE, EDWARD    
GRO Reference: 1851  D Quarter in CARLISLE  Volume 25  Page 68
MCKINZIE, WILLIAM    
GRO Reference: 1854  S Quarter in COCKERMOUTH  Volume 10B  Page 438
MCKINZIE, JAMES    
GRO Reference: 1857  S Quarter in COCKERMOUTH  Volume 10B  Page 435
MCKINZEY, ELIZABETH  ANN  
GRO Reference: 1862  S Quarter in COCKERMOUTH  Volume 10B  Page 495
MCKENZIE, ANNIE  ELLEN  
GRO Reference: 1872  J Quarter in WIGTON  Volume 10B  Page 498
By 1881 William senior is a widower. Daughter Ann on census
MCKINZIE, JEMIMA    died age 49  
GRO Reference: 1877  S Quarter in COCKERMOUTH  Volume 10B  Page 313
Colin