I've found John Beck's baptism, mistranscribed as Back:
IGI Individual Record FamilySearch™ International Genealogical Index v5.0
British Isles
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JOHN BACK Pedigree
Male
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Event(s):
Birth:
Christening: 19 NOV 1810 Braddan, Isle Of Man, England
Death:
Burial:
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Parents:
Father: JOHN BACK Family
Mother: MARY HASTINGS
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Messages:
Extracted birth or christening record for the locality listed in the record. The source records are usually arranged chronologically by the birth or christening date
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There's a tree on Ancestry which has John Beck and Mary Hastings as the parents of John Beck, but doesn't have his baptism --- in fact, it gives his birth year as circa 1815. This tree is from a g g g granddaughter of John's brother, Henry Beck bap 1819:
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HENRY BECK Pedigree
Male
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Event(s):
Birth:
Christening: 13 JUN 1819 Braddan, Isle Of Man, England
Death:
Burial:
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Parents:
Father: JOHN BECK Family
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Messages:
Extracted birth or christening record for the locality listed in the record. The source records are usually arranged chronologically by the birth or christening date
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John Beck b 1810 left a will. Have you seen it, Peter ?
BECK, John 1865 Lonan E 87 0106467
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This might be the will of John Beck's mother, Mary nee Hastings ?
BECK, Mary 1836 Braddan E 0106436
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I have no idea whether the Elizabeth Beck below, b circa 1728, is connected to the family or not (maybe a grandmother of John b 1810 ?) ---burial:
BECK, Elizabeth 86 Braddan 16 Mar 1814 DICKENSON
It would be worth seeing this will just in case there's a connection:
BECK, Elizabeth 1814 Braddan A 19 0106239
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The tree gives the following details for the parents of John Beck b 1810:
Father John Beck b circa 1777 co Down, Ireland. Died 1856 Douglas (but there's no burial listed in Brian Lawson's index). In 1841 he's a 60 yr old labourer from Ireland in Quilliam Court, Douglas (Conchan 7/17). In 1851 he's a 74 yr old servant / farm labourer in the District House of Industry, Douglas, b Co Down, so this is probably where the tree compiler got her birth year and place of birth from.
Mother ---Mary Hastings. It says "b about 1790 in Ireland ?"
Hope this helps.
Jean C