The problem here isn't that energy needs to be defined more rigorously like everything else. The problem is that you're making an incorrect assumption that everything else can be rigorously defined for once and for all. For example:
"[...]using our intuition we know what momentum should be and also we know that defining it as p=mv is a good definition."
Actually this doesn't work. For example, a beam of light has zero mass and nonzero momentum, so p=mv is false for light. If intuition told you p=mv, intuition was wrong.
The general way to go about defining a conserved quantity is to pick something that is your standard amount of that quantity, and then use experiments to find out how much of various things can be converted into that standard. For example, if you pick a 1.00 kg mass moving at 1.00 m/s as your definition of the unit of momentum, then you will find through experiments that its momentum can be exchanged for 2.00 m/s worth of motion for a 0.50 kg mass. This naturally leads to the hypothesis that p=mv. Further experiments seem to verify that hypothesis. But then eventually you do experiments with electrons moving at 30% of the speed of light, or with beams of light, and you find out that p=mv was wrong. It was only an approximation valid under some circumstances. You're forced to revise your definition of p. It's a purely empirical process.
Same thing for energy. The only approach that fundamentally works is to define something as your standard unit of energy. This could be the energy required to heat 0.24 g of water by 1 degree C. Then experiments would show that you could trade that amount of energy for the kinetic energy of a 2.00 kg object moving at 1.00 m/s. Ultimately, all you can do is proceed empirically.
"[...]recently I've heard that on general relativity there's a loss of some conservation laws [...]"
Yes, and this is why I don't agree with Jerry Schirmer's answer. He says that energy is the conserved quantity that you get because of time-translation invariance. But this procedure doesn't work in GR. In technical terms, the relevant symmetry becomes diffeomorphism invariance, and that doesn't satisfy the requirements of Noether's theorem. The more fundamental reason it can't work in GR is that in GR, energy-momentum is a vector, not a scalar, and you can't have global conservation of a vector in GR, because parallel transport of vectors in GR is path-dependent and therefore ambiguous. What you can do in GR is define local (not global) conservation of energy-momentum. Even if the technical details are mysterious, I think this counterexample shows although Noether's theorem does provide a deeper insight into where conservation laws come from, the ultimate definition of conserved quantities is still empirical.
BTW, there is a good exposition of this philosophical position in the Feynman Lectures. He discusses conservation of energy using the metaphor of a bishop moving on a chess board and always staying on the same color. Although that treatment is aimed at people who don't know anything about Noether's theorem or general relativity, I think his philosophical position holds up very well in the full context of what is currently known about all of physics.