There is nothing wrong with calling the “Agnus Dei”, the “Agnus Dei”. Or if using English, “Lamb of God”. Church documents use this as the name of the prayer. “Ecce Agnus Dei” would be required when the priest prays the prayer. And this prayer has never been allowed to be ad libbed.
Here is the Church using the name “Agnus Dei”. There are other documents as well, and Agnus Dei is used.
Redemptionis Sacramentum
On certain matters to be observed or to be avoided
regarding the Most Holy Eucharist
[73.] In the celebration of Holy Mass the breaking of the Eucharistic Bread – done only by the Priest celebrant, if necessary with the help of a Deacon or of a concelebrant – begins after the exchange of peace, while the Agnus Dei is being recited. For the gesture of breaking bread “carried out by Christ at the Last Supper, which in apostolic times gave the whole eucharistic action its name, signifies that the faithful, though they are many, are made one Body in the communion of the one Bread of Life who is Christ, who died and rose for the world’s salvation” (cf. 1 Cor 10,17).[153] For this reason the rite must be carried out with great reverence.[154] Even so, it should be brief. The abuse that has prevailed in some places, by which this rite is unnecessarily prolonged and given undue emphasis, with laypersons also helping in contradiction to the norms, should be corrected with all haste.[155]