Energy is a measure that is associated with a state (or condition)of one or more bodies. A change in thermal energy is demonstrated by a change in the temperature of a body.
When Vasili Alexeev lifted 562 lb above his head, he did not increase the weights kinetic energy. But he did increase the separation between the weights and the Earth, which attracts each other via the gravitational force. His work increased the gravitational potential energy of the weights-Earth system by changing its configuration, that is, by shifting the relative locations of the Earth and the weights. Gravitational potential energy is the energy associated with the state of separation between bodies that attract each other via the gravitational force.
If you compress or extend a spring, you change the relative locations of the coils within the spring. They resist that reconfiguration, and the result of your work is an increase in the elastic potential energy of the spring. Elastic potential energy is the energy associated with the state of compression or extension of an elastic (springlike) object.

In the language that describes energy, potential energy is often said to be “stored” in a system, in the sense that it could later result in motion.
When each stone of the Rheims cathedral was shifted from a quarry to its present height potential energy was “stored” in the stone-Earth system. However, using less precise language, you might sat that energy is stored in the stone itself, and you might refer to “the potential energy of the stone” or even that of the cathedral as a whole. Such language is often acceptable, but you should keep in mind what is actually meant: the Earth is an important part of the energy-storing system.