My source shows five John Cartwrights among those who served in War of 1812.
(1) One served as Private in Green's Regiment, Mounted Infantry, Virginia Militia. (2) Another served as a Private in Major Lillington's Detachment, North Carolina Militia (3) A third served as a Private in the 4th Regiment (Morgan's), Louisiana Militia (4) The fourth served as a Private in the 27th Regiment (Long's), Maryland Militia (5) The fifth served as a Private in the 32nd Regiment (Hood's), Maryland Militia
Any two or more "might" have been the same person. No pension claim appears to have been filed for your John Cartwright.