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There are two foundative Equivalence Principles in General relativity:

Weak Equivalence Principle (WEP): the dynamics of a test particle in a gravitational field is independent of its mass and internal composition. (WEP is equivalent to say that the ratio between the gravitational mass $m_g$ and the inertial mass $m_i$ has a universal value $k$. This value can be considered as the scalar $1$).

Einstein equivalence principle (EEP): A frame linearly accelerated relative to an inertial frame in special relativity is LOCALLY identical to a frame at rest in a gravitational field.

Most textbooks say that "obviously" (EEP) implies (WEP) but the converse is not true.

I don't understand why the implication (EEP) $\Rightarrow$ (WEP) is true, and moreover I'd like a counterexample showing that (WEP) $\not\Rightarrow$ (EEP).

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EEP has something more than what WEP has. WEP states that in a small reign of space-time,there is no difference for a particle to free fall (to move in a gravitational field) or to move in a box with acceleration $g$ in the inverse direction.

Then Einstein stated more and said that not only for free fall but also for any non-gravitational experiment, this equivalence exists.

For example, we consider an electron and a proton.They fall with the same acceleration in the gravitational field and independent of their mass.If we then combine them and make an atom,the free fall acceleration of that atom would be the same.We know that the mass of atom is less than the total mass of electron and proton and negative potential is added to the system. Here we see that in addition to mass,gravitation is also coupled with energy.

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I agree that in the textbook you considered, explanation is not clear. First, please check my answer here: graviton and principle of equivalence

Personally, I prefer to unify WEP and EEP, but let's distinguish them as in your textbook. The example you provided for WEP is definitely mechanical, so not-gravitational experiment, hence the implication (EEP) ⇒ (WEP) indeed is obvious. As counter example I would suggest an experiment with photon. For instance the Doppler effect, being a consequence of special relativity is a part of general relativity. Being an experiment on photons, it does not tell you anything about mass of particles.

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