Looking at the census etc. these appear to be 2 different men. The 1851 man was a coal miner and remained one, remarrying (aged 30 which would be correct) an Eleanor Robinson 20/1/1855 Auckland St. Helen, Durham. He is shown as son of a Ralph (which would explain why his first child had this name).
In 1861 he is living in Evenwood Barony, Durham, still a coal miner, born Durham, but for some reason his age shown as 30, and his wife listed as Esther (when it should be Ellen). With them is his first born son Ralph plus a son William bc.1852 Houghton le Spring, stepson Thomas W. Robinson bc.1855 Bp. Auckland and a son Anthony bc.1857 Durham.
In 1871 his age is once again right, still a coal miner, wife Ellen, prev. mentioned children Ralph, William, Thomas (now shown as Turnbull instead of Robinson), Anthony plus 4 more children born between 1861 and 1871.
So really, which Anthony is it you are interested in - the one who was a Stone Quarryman married to Isabella in 1861 census or the one first married to Barbara? By the way, she didn't die in 1851, that one was aged 75! She appears to have died aged 27 in Sept.qtr.1854 Durham which would tie in with him marrying Eleanor (Ellen) Robinson the following year.
The 1861 Anthony appears to have married Isabella Robson 26/12/1852 Crook, Durham. (per FindMyPast) but unfortunately doesn't state their fathers names so can't identify - except on Ancestry there is a tree that states his parents as Edward Turnbull and Elizabeth Ann Kell. Full details of the marriage do not appear to be online, just the date and place, so obtaining the marriage certificate (maybe a Register Office wedding) would confirm if this Anthony was indeed the son of an Edward. His first child with Isabella was a son Edward in 1854, who died the following year, and GRO birth index on their own site now lists mothers maiden names and confirms his mother was Robson. This Anthony died 3/2/1866.
So, from what I can see, both men born around the same time but one was son of a Ralph and the other likely to have been son of an Edward.
Annette