Probably a language thing, but "Eucharist" tends to refer to the actual Liturgy in the Greek Church, since "Eucharist" still literally means "Thanksgiving" in Greek.. Therefore, in Greek practice, when the Eucharistic Prayer is introduced (as I understand that it is still done in the RC Church), the dialogue goes:
Priest: Lift up your hearts.
People: We have lifted them up to the Lord.
Priest: Let us Eucharist (i.e. give thanks to) the Lord;
People: It is meet and right.
....and then begins the Eucharistic Prayer.
So the Liturgical Act of offering the Eucharist, tends to be called "Eucharist", since, linguistically, it is "the Lord" who is "Eucharist-ed"; and our receiving of the Gifts tends to be called "Communion".