That looks good - that those shipping records agree with the Cheshire records.
Cheshire Birth indexes for the years: 1864.
Surname, Forename(s), Sub-District, Registers At, Reference:
CAIN, Bernard, Runcorn, Halton, RUN/26/29

www.cheshirebmd.org.uk

There is also the other Bernard Cain 1895, b.New York (1911 census).

Sue