I have looked at all the US records I could find, and I have the following:
John Harrison birth 12 September 1804 (based on death record which gives his age in years-months-days).
Charles Harrison was 47 years old at the time of death (from gravestone) which puts his birth between 11 May 1802 and 10 May 1803.
Their place of birth as Coleraine comes from the death record of Charles' son William which states that both William and his parents Charles Harrison and Jane Graham were born in "Colraine, Ireland". William's obit states that he was born in "Colraine, County Farmanagh Ireland". His sister Jane's obit says she was born in "Colrain, Ireland". I assume that the reference to Fermanagh is a mistake in the obit, since I have not been able to locate any place in Fermanagh called Colraine.
I also have the exact birth dates of all Charles' children (all of whom were born in Ireland, presumably Coleraine): John - 12 March 1829 Isabella - 3 March 1831 Margaret 8 Dec 1833 Sarah - 5 Sep 1836 David - 17 Jun 1839 William - 2 Feb 1843 Jane - 4 Feb 1846
Since I live in the US, I won't be able to get to research the PRONI records any time soon. I have looked at the Charles in rootsweb, but nothing to link that Charles to my ancestor. Are there other on-line records (other than those available through Ancestry.com or Rootsweb that you can recommend ?