where can one find the talmud text or halachah text or something similar that identifies or divides Jews as either Ashkenazim or Sephardim or Teymanim? (Also, is it two only, the first two only, because the last one, Yemenites or Teymanim, is sometimes left out)?
There was the Yeshiva in France, of the Tosafists, and there was a chain developing independently in Spain. These two schools had some differences in approach and in Minhag. After the Spanish expulsion, when many Spanish Jews moved East and carried the title Sfardy along with them, the divide became more pronounced.