Speculation has gotten me into trouble before, but here are the Collins families in Jefferson County NY, 1820. Without more evidence, we won't know which, if any, of these are related to Stephen L. Collins. High on my list will be checking for will/probate records. Also, I have not yet determined if any of these 5, were born in Vermont. Only three of these households had a male under the age of 10 years. If S.L. was born after the census (official date for the enumeration was August 7th, 1820), then he would not have been counted. Most records indicate that S.L. was born before that date (most likely in May). Here are all 5 families.
Collins, Clarissa, Henderson Twp: 1 male (0-10), 1 female (0-10), 1 female (16-26)
John Collins, Watertown: 1 male(16-26), 1 male (25-45), 1 male (45+), 3 females (0-10), 3 females (10-16), 1 female(26-45), 1 female (45+)
John B. Collins, Ellisburg: 4 males (0-10), 2 males (10-16), 1 male (26-45), 1 female (0-10), 1 female (10-16), 1 female 26-45)
William Collins, Ellisburg: 1 male (45+), 1 female (0-10), 1 female (16-26), 1 female (45+)
William Collins Jun, Ellisburg: 2 males(0-10), 1 male (26-45), 1 female (16-26)