Relationship between intutions on energy

Hi all,

After reading Feynman's Lectures I see that his way to understand energy is as a conserved quantity. On the last days then I've seem Susskind's lectures on classical mechanics and he also defines energy like that. He says that energy is a conserved quantity, being able to characterize groups of states of a system.

So the subsets of the set of all possible states are each of them characterized by a certain amount of energy. That's all fine and it gives a nice understanding that the main point of energy is it's conservation.

I'm still struggling however to relate this intuition on energy with the most common ones. People usually see energy as capability of a system on doing things. Even the word energy as I google means "activity". And also, there are those intuitions like: "a particle in the gravitational field of earth needs to receive external energy to be able to be lifted and overcome gravity".

How all of those intuitions on energy agree with the definition of energy simply as a conserved quantity? I know how to get the formulas, put then together and show that the total energy is conserved, that's not my doubt. My doubt is really on the intuition behind.

Thanks very much in advance.