In the 1851 census the family is listed as Morison, the father is Alexander and there is a Alexander and John listed as sons.
Do not read anything into spelling variations.
Jessie is born in 1869 and in the 1871 census there is a Jessie Morrison (born in Marnoch abt 1870)
No, there is not. The original census
never gives a date of birth. There is a Jessie Morrison,
aged 1. This is exactly accurate because if she was born on 17 November 1869, she had passed her first birthday on 17 November 1870, but on the date of the 1871 census, 2 April, she had not yet passed her second birthday. So her age is indeed one year.
The same applies in 1881. She passed her 11th birthday on 17 November 1880, but on census day, 4 April 1881, she had not had her 12th birthday, so was correctly listed as aged 11.
Some census transcriptions (mis)calculate people's supposed year of birth by subtracting their stated age from the census year. As the census was usually taken a quarter of the way through the year, three-quarters of people had not had a birthday in that year, so the subtraction method gives the wrong answer three-quarters of the time.
I don't think there can be much doubt that you have the right Jessie Morrison. The dates, places and parents' names all match. What address did her father state on Jessie's birth certificate?
I have researched the records at Ancestry.com and some at Scotlandspeople.gov.uk
See
http://www.rootschat.com/forum/index.php?topic=714261.0