I believe the American BY system was started by Reb Viche Kaplan. She was one of the closest of Sara Schnirer’s talmidos. She married R’ Boruch Kaplan (a talmid of R’ Yaakov Yosef Herman of All for the Boss). He learned overseas and eventually a shidduch was made. They came to America and she started a school. A lot of the story is told in one of the books by Rebbetzin Shain, possibly All for the Boss or one of her later ones. The basic BY hashkofo was to adhere to Torah and keep the mitzvos, strengthening a girl’s emuna and bitachon, teaching chumash and novi and even mussar seforim, giving her the tools to build a Jewish home, which the secular world was pulling girls away from right and left. These girls left the BY system understanding the value of a husband who learns Torah.
I believe the differences in hashkofos only came about after there were enough students for individual groups to open their own schools, such as different chassidim and such. They each added their own flavor to the basic BY hashkofos. Many immigrants came to the US before and then after the war, bringing with them the traditions of their home towns, adding even more spice to the melting pot that was America in those days. The seminaries in EY simply reflect those differences.