Inside Boston Latin School on a recent Saturday, educators presented a diverse crowd of fifth-graders a golden opportunity: admission to a free, two-week summer test-preparation program that could help them get into one of the city’s prestigious exam schools. For the second year in a row, in the hopes of boosting diversity, school officials nearly doubled the number of seats in the program, inviting not just those with the top standardized test scores but also promising students from forgotten corners of the city.