In the 'open door' example above, I possess potential chemical energy (energy stored in me), and by lifting my hands to push the door, my action also had kinetic energy (energy in the motion of my hands). By pushing the door, my potential and kinetic energy was transferred into mechanical energy, which caused work to be done (door opened). Here ...
Elastic energy is the potential mechanical energy stored in the configuration of a material or physical system as work is performed to distort its volume or shape.  Elastic energy occurs when objects are compressed and stretched, or generally deformed in any manner.
Mechanical energy is the sum of kinetic and potential energy in an object that is used to do work. In other words, it is energy in an object due to its motion or position, or both. In the 'open door' example above, I possess potential chemical energy (energy stored in me), and by lifting my hands to push the door, my action also had kinetic energy (energy in the motion of my hands).
In physics, mechanical energy (E mech) is the energy associated with the motion and position of an object usually in some force field (e.g. gravitational field). Mechanical energy (and also the thermal energy) can be separated into two categories, transient and stored.
Mechanical energy and thermal energy are two forms of energy. These concepts are very critical in fields such as mechanical systems, heat engines, thermodynamics and even biology. It is critical to have a clear understanding in these two concepts to master these fields.