"You shall be the enemy of heaviness, even in spirit.
"That makes you light for your horse.
"When your heart is light,
"so is your hand.
"When your heart is light,
"it drives you forward.
"Nothing drives the melancholy,
"the heavy-hearted forward.
"Forward, however, is everything."
Rudolf Binding (Reitvorschrift für eine Geliebte, München, w.o. year, 13, translation: TR).
Energy is the prerequisite for impulsion, almost like a raw material. As many significant terms, it has a psychological as well as a physical aspect. In the psychological dimension, energy is the horse's readiness to use all his strength in support of the rider. You could consider it the horse's commitment to the task at hand. In lessons, I often tell my students that the horse has to make a bigger effort in order to increase the impulsion and improve the performance. That means the horse has to increase his energy output, which begins with the horse's mind and heart, before it translates into a physical manifestation.
In the physical dimension, energy refers to a certain muscle tone, a certain "vibratory rate", if that makes sense. Alexis L'Hotte refers to it as "vibrating haunches". The horse's muscles have to feel as if they were charged with electricity, just waiting for the slightest indication of the rider to discharge their stored up energy. This is what you experience, e.g., when you release the horse from the collected trot into the extended trot, or when the slightest touch of your calf at the halt triggers the piaffe.
Engagement seems to be a typically English term. I can't think of a true German equivalent, and I am not familiar enough with all the French technical terms. My impression is that the term Engagement is used differently by different people. I personally have come to define it as the degree to which the hind leg approaches the center of gravity, i.e. how far the horse steps under. Other riders will possibly define it differently, usually in a way that overlaps with energy, impulsion, and possibly collection.
Unlike energy and engagement, Impulsion ("Schwung" in German) is a member of the original training pyramid (Skala der Ausbildung), as formulated in the German Army Riding Instruction from 1912 (Heeresdienstvorschrift 1912) and its successors all the way to the "Richtlinien für Reiten und Fahren" that are issued by the German FN. In my understanding, impulsion is a composite concept, rather than an elementary one. I consider it composite, because it is not something you can ask for in a direct way, especially on relatively untrained horses. It is a product of several other, more basic elements, that need to be in place before impulsion can manifest itself. Energy and thrust can already be there, but impulsion is a more refined form of energy.
The main prerequisites of impulsion, in addition to energy and thrust, are relaxation, straightness, and balance: Balance, because only when all four legs carry the same amount of weight can the rider regulate the horse's carrying and thrusting forces as well as their ratio to one another with the necessary precision. Balance implies a certain degree of flexion of the haunches, which is necessary for the rider to determine the direction of the thrust of the hind legs. The more the rider can direct the thrust in a vertical direction, the springier the gait will become, i.e. the more impulsion will be created.
Relaxation is necessary, because only a relaxed body and mind allow the energy created by the hind legs to sweep through the entire horse from the hind hooves to the mouth and back. Tensions, on the other hand, act as road blocks that swallow most of that energy and allow only a tiny fraction to pass through, which leads most riders to intensify their aids instead of removing the block.
From a slightly different angle, only a relaxed body enables the horse to use all his joints with the maximum range of motion. Any stiffness automatically reduces the range of motion. Impulsion is unthinkable with anything less than the maximum freedom of movement of all participating joints.
Straightness is required, because only when the forces of the hind legs are directed immediately towards the center of gravity is there no loss of energy, and they can act as true springs that are compressed and released in a regular rhythm, in which the release itself provides the necessary momentum to flex the spring again in the next stride: the horse becomes a perpetuum mobile. When relaxation, balance, straightness, as well as cultivated energy come together, the gait will increase in elasticity and expressiveness. Impulsion and cadence are born. You could say that impulsion is the ultimate refinement of the hind legs' energetic thrust, while cadence is the refinement of the regularity of the rhythm. Together, impulsion and cadence create brilliance. Since the straight and balanced horse flexes his haunches in proportion to the amount of thrust, increasing the impulsion will also indirectly increase the collection.