How can we define energy other than the definition that it's a capability to do work? It is actually a property of energy that it can do some work not an actual mean to define it because we cannot define a thing on the basis of what it is doing or what it can do.
 A: To set up a mathematical model that describes a physical system, one has to define observables, and one has to define "laws", i.e. axioms imposed on the mathematical model so that there is a correspondence of the mathematical solutions to the measured data and also predictions for future behavior.
Here we start with the definition of a force in words:

One of the foundation concepts of physics, a force may be thought of as any influence which tends to change the motion of an object.

"Foundation" means it is a definition, expressed differently in different physics models

This leads sequentially to  to the concepts of "work" "energy" "power"

work:refers to an activity involving a force and movement in the directon of the force. A force of 20 newtons pushing an object 5 meters in the direction of the force does 100 joules of work.
energy:is the capacity for doing work. You must have energy to accomplish work - it is like the "currency" for performing work. To do 100 joules of work, you must expend 100 joules of energy.

So in the manner that the physics models have developed, there is no other definition.
Can one invent another definition? Yes, Example :One may start postulating the Lorenz transformations on four vectors  and postulate that "energy is the value in the four vector that gives as the length the invariant mass of the particle given the momentum". Momentum, mv, is the basis in this case. This would be correct, but a very complicated way of defining energy, particularly energy in the emergent classical mechanics system.
A: "[…] we cannot define a thing on the basis of what it is doing or what it can do."
Why not? Try defining a progressive wave other than in terms of what it's doing!
In my opinion "The energy of a system is the amount of work it can do" is an excellent starting definition of energy. It enables one to derive the Newtonian formula ($\frac{1}{2}mv^2$) for kinetic energy, and formulae for potential energy in uniform and inverse square law fields, escape velocities, closest distances of approach and so on.
Later on one discovers various difficulties with the "amount of work it can do" definition. The Second law of Thermodynamics is, on the face of it, a glaring example. If heat is a form of energy, but you can't turn it all into work, that seems to generate a contradiction. [This argument is (deliberately) rather sloppily stated.] Another example: according to quantum mechanics an oscillatory system (e.g. a diatomic molecule) has a zero point energy, that is a minimum energy that cannot be removed from the system, so the system's energy is greater than the amount of work it can do!
There are ways of getting round these difficulties and preserving the "amount of work" definition of energy, but they come at the cost of caveats and interpretative devices (arguably sophistry) that spoil the original simplicity of the idea. In fact one may wish to abandon "the amount of work it can do" as a definition of energy, and to look upon energy as a conserved quantity which can be calculated for various systems by specific formulae or equations. An admission of defeat?
A: From my perspective, it is just a number associated with a system or object. For instance, a 2 kg ball moving with velocity 5 m/s has kinetic energy of 25 J. This number is always conserved for a system of objects (for instance 2 balls colliding, or two charges acting on each other) provided that no external forces act on it.
