Characterization of some binary words with few squares

Thue proved that the factors occurring infinitely many times in square-free words over {0,1,2}{0,1,2} avoiding the factors in {010,212}{010,212} are the factors of the fixed point of the morphism 0↦0120↦012, 1↦021↦02, 2↦12↦1. He similarly characterized square-free words avoiding {010,020}{010,020} and {121,212}{121,212} as the factors of two morphic words. In this paper, we exhibit smaller morphisms to define these two square-free morphic words and we give such characterizations for six types of binary words containing few distinct squares.