Hi Billy
I did a search on FreeBMD and found an Elsie Annie GREEN born 1893 Birmingham. The details are :-
GREEN Elsie Annie Birmingham June Quarter 1893 6d 91.
I then went to the 1901 Census but couldn't find an Elsie born 1893 ish to a Jonah in Warwickshire.
I think, given the evidence, that you have the right Elsie despite your relatives saying her birth was Jan not Feb. It is highly likely that Elsie herself may not have been too sure of the month. The birth certificate is more likely to be right I am afraid.
Cheers
Karenlee