I've scraped together some more information and I think Captain Alexander is Provost John's son.
According to a pamphlet published in 1917 by the Niagara Historical Society, John Dickson of Conheath had four daughters and six sons, and it names the sons: Robert, William, John, Alexander, Thomas and Walter. William, Thomas and Robert went to Canada and did well (Robert married the daughter of a Sioux chief). Walter became a lawyer in Edinburgh and married a Goldie. I've also found three of the daughters. Here are the dates I know:
John James Dickson marries Helen Wight (1741-1784) in March 1758. Children:
Elizabeth, d. 1825
Margaret, d. 1827
Robert, 1768-1823
William, 1769-1846
Thomas, 1775-1825
Walter, 1776-1855
No birth date for Alexander, and nothing at all for John and one unnamed daughter.
My conjecture is that Alexander is the first child, born in or around 1759, which would allow him to reach a plausible age - just - to be a captain lieutenant in the Royal Artillery in 1777.
On the other hand, John James Dickson apparently had two brothers, Thomas and Nicholas, about whom I can find nothing. Alexander could be one of their sons, born slightly earlier...