I'm having trouble understanding the differences between the different forms of the Equivalence Principle. I've search in other questions and I think that this was not previously asked, at least in the particular way I'm trying to say it.
1) the weak equivalence principle states that the inertial mass is equal to the gravitational mass. Equivalently, it states that the path followed by a particle in a gravitational field only depends on initial position and velocity and not on any particular magnitude it has (like mass, charge, etc.).
2) The Einsten Equivalence Principle states that the weak equivalence principle is true AND that any non-gravitational experiment held on a free falling system will give the same results as in a rest frame. Equivalently, it states that experiments in a accelerated frame and a frame under the influence of gravity are indistinguishable.
3) The Strong Equivalence Principle is exactly like the Einstein Equivalence Principle but it applies to every kind of experiment, including gravitational ones.
First of all, are these the right definitions? I mixed different statements in different websites and books to make it more clear (to me).
Secondly, I'm having trouble seeing why it is so important for the Strong Principle to include gravitational experiments. What could be an example of an experiment violating the Strong Principle but not the Einstein Principle?