Concerning the change in terminology by Eugene Nida (inventor of the method), here is what Nida wrote:
"Unfortunately, the expression 'dynamic equivalence' has often been misunderstood as referring to anything which might have special impact and appeal for receptors. Some Bible translators have seriously violated the principle of dynamic equivalence as described in Theory and Practice of Translating and Toward a Science of Translating. It is hoped, therefore, that the use of the expression 'functional equivalence' may serve to highlight the communicative functions of translating and to avoid misunderstanding" (Eugene Nida and Jan de Waard, From One Language to Another (Nashville: Thomas Nelson, 1986, vii-viii).
So Nida would not consider The Message and other paraphrases to be DE/FE. In fact, some were saying such things, and that is why Nida changed his term.