Understanding the origins of the diversity in terrestrial planets is a fundamental goal in Earth and planetary sciences. In the Solar System, Venus has a similar size and bulk composition to those of Earth, but it lacks water. Because a richer variety of exoplanets is expected to be discovered, prediction of their atmospheres and surface environments requires a general framework for planetary evolution. Here we show that terrestrial planets can be divided into two distinct types on the basis of their evolutionary history during solidification from the initially hot molten state expected from the standard formation model. Even if, apart from their orbits, they were identical just after formation, the solidified planets can have different characteristics. A type I planet, which is formed beyond a certain critical distance from the host star, solidifies within several million years. If the planet acquires water during formation, most of this water is retained and forms the earliest oceans. In contrast, on a type II planet, which is formed inside the critical distance, a magma ocean can be sustained for longer, even with a larger initial amount of water. Its duration could be as long as 100 million years if the planet is formed together with a mass of water comparable to the total inventory of the modern Earth. Hydrodynamic escape desiccates type II planets during the slow solidification process. Although Earth is categorized as type I, it is not clear which type Venus is because its orbital distance is close to the critical distance. However, because the dryness of the surface and mantle predicted for type II planets is consistent with the characteristics of Venus, it may be representative of type II planets. Also, future observations may have a chance to detect not only terrestrial exoplanets covered with water ocean but also those covered with magma ocean around a young star.