A positioning tool can include a mandrel and an engagement device reciprocably disposed on the mandrel. The engagement device can include at least one engagement member and at least one landing dog. The mandrel can displace relative to the engagement device in response to engagement between the engagement member and at least one internal profile in an outer tubular string, and the landing dog can extend outward in response to displacement of the mandrel in a selected longitudinal direction relative to the engagement device. A system for use in a well can include a tubular string and a positioning tool reciprocably disposed in the tubular string. The positioning tool can include a landing dog that extends outward from a retracted position to engage one or more internal profiles of the tubular string, in response to a pattern of reciprocation of the positioning tool in the tubular string.