Note that Jane hasn't just had one Birkin, just one at a time. In her lifetime since the mid-80s she has had quite a few, many have been sold to raise funds for charities through the years. There is no the Jane Birkin's Birkin, there is only a Jane Birkin Birkin.
I love JB and her amazing style BUT...She has weirdly told many different stories about her now infamous meeting with Jean-Louis Dumas in the early 1980s, each interview slightly different. Much of embellished. She once said she drew the bag mid-flight on a sick bag, another on a napkin, sometimes it was him, when we. all know the Birkin is just an HAC 'elan', the HAC was already nearly 100 years old by the time she 'coincidentally' bounced into J-L D. They also knew who each other were, both Serge and her had been very good Hermes clients, Serge having a travel Kelly he took everywhere when on the move. By 1981 she was no longer the ingenue she made herself out to be or made her famous, she was totally established and already a style icon. Both parties knew exactly what they were doing.
I don't think 'we do all know'
a lot of people think the HAC is a variant of the Birkin rather than vic-versa
As you point out, it is interesting to observe that the the bag 'designed' for Jane Birkin
was not as original as many people might think