Consider some popular formulations of the energy-time relation.
(1) When measuring an energy of a system, the accuracy of the measurement cannot exceed h/t, where t is the duration of the measurement.
(2) When preparing a system in a particular state, the uncertainty of the energy of this state will be at least h/t, where t is the preparation time, and h is the Planck constant.
These two formulations are essentially equivalent, since measuring the energy of the system is the same as preparing a state where the energy has a definite value. I think Landau-Lifshits textbook states (1) as a viable formulation of the energy-time uncertainty relation. But as shown by Aharonov and Bohm, (1) (and therefore (2)) are incorrect. It is possible to set-up a non-demolition measurement of energy which takes an arbitrarily short time and has an arbitrarily good accuracy.
(3) If some property of a system changes substantially on a time scale t, then the energy of the state has uncertainty at least h/t.
A version of this was stated by Bohr and Wigner. This is the formulation which "explains" why an unstable particle (resonance) does not have a definite energy. It is a bit hard to make this principle precise, and in fact there are many slightly different formulations. But it can be proved rigorously.
(4) If an internal (dynamical) clock of a system has accuracy t, then the energy of the system is uncertain, with uncertainty being at least h/t.
This is more or less equivalent to (3).
There is a well known story (told, for example, in R. Peierls's wonderful book "Surprises in theoretical physics") about Einstein inventing a counter-example to (1), and Bohr refuting him using Einstein's own General Relativity Theory. In retrospect, Bohr's refutal, while correct, seems beside the point, since (1) is not true in general.