The proper term is "Extra-ordinary Ministers of the Eucharist." Therefore, it would be synonymous to say, "Extra-ordinary Eucharistic Ministrers." I think the inclusion of the word, "extra-ordinary" is an important reminder that the role should not be allowed to become ordinary, as it often has. (Sometimes, I think some EMEs think they're just doing an extra-ordinary job!) But an "extraordinary Minister of the Eucharist," is, nonetheless, a kind of minister of the Eucharist.
The proper term is "Extraordinary ministers of Holy Communion" not "Extraordinary ministers of the Eucharist."