ANSWER: the "low bar" shape was invented to allow women with long dresses and skirts to ride a bike more easily and to mount it without "being indecent". This also agrees with what I was taught as a child.
However no one has provided any links to these explanations, yet. I'll fill the gap by providing some useful links to the history of the bicycle that cover the OP question.
Evidence:
These article all confirm what I and other have said about the origin of the different shape, but also point out that this allowed a real costume revolution at the turn of 20th century, empowering women to become more independent.
