We discuss how knowledge of the whole evolutionary history of dwarf galaxies, including details on the early star formation events, can provide insight on the origin of the different dwarf galaxy types. We suggest that these types may be imprinted by the early conditions of formation rather than only being the result of a recent morphological transformation driven by environmental effects. We present precise star formation histories of a sample of Local Group dwarf galaxies, derived from color-magnitude diagrams reaching the oldest main-sequence turnoffs. We argue that these galaxies can be assigned to two basic types: fast dwarfs that started their evolution with a dominant and short star formation event and slow dwarfs that formed a small fraction of their stars early and have continued forming stars until the present time (or almost). These two different evolutionary paths do not map directly onto the present-day morphology (dwarf spheroidal versus dwarf irregular). Slow and fast dwarfs also differ in their inferred past location relative to the Milky Way and/or M31, which hints that slow dwarfs were generally assembled in lower-density environments than fast dwarfs. We propose that the distinction between a fast and slow dwarf galaxy primarily reflects the characteristic density of the environment where they form. At a later stage, interaction with a large host galaxy may play a role in the final gas removal and ultimate termination of star formation.