Yes, too many Bernards. I agree with you that the key cert would be:
Cheshire Birth indexes for the years: 1864.
Surname, Forename(s), Sub-District, Registers At, Reference:
CAIN, Bernard, Runcorn, Halton, RUN/26/29
[for Bernard the father].

And also Carol's suggestion of their son Bernard Cain's birth registration March quarter 1895: Cain, Bernard, Liverpool Vol 8b, Pg 35, to see whether it says anything about being born in New York, USA.
All their other children were born in Liverpool, according to the 1911 census signed by Bridget Cain als Doherty, including Margaret who was 2 years older than Bernard.

Sue