Flatbush was the town--there was no town of Midwood, but even back to the Dutch era, Flatbush and Midwood (or their Dutch origin names) were synonymous.
From my memory and usage I saw in the '50s (from people who lived there back to the 'teens or earlier) Flatbush was generally regarded as the area more or less bounded by a line through Prospect Park on the north through to Empire Boulevard (but this was also regarded to sort of include Ebbets Field), Nostrand Avenue on the east, the LIRR on the south and Coney Island Avenue on the west.
South of this with the same east-west boundaries was Midwood down to about Kings Highway.
Out on Long Island a lot of the names of unincorporated areas often follow the school district. If you went to Erasmus you were in Flatbush. If you went to Midwood High...