For a 'person'(don't get offended) to be called Avatara, he/she has to be born on the our Earth in material form experience Earth-ness but without loosing any divine power.
The word, Avatar itself mean to descend as saviour as अव- (ava-, “down”) + तार (tāra, “savior”).Wikitionary
By definition an incarnation is different. It requires the birth of God in physical form, through the natural process and his existence upon earth in physical form undergoing the same experiences as living beings. It is not necessary that the physical form needs to be that of human always. It can be a human, animal or semi human or even mythical form. The incarnation may exist for a brief period of time or for a life time. The incarnation may have all the powers and awareness of God in His absolute aspect or only a necessary portion of it.
Bhagavata Purana do list Mohini as an Avatar of Vishnu in the 26 Avatars, but also comments on it being infinite. It was Garuda Purana which counts the Avatars to be 10.