This question makes me recall my childhood. I see two approaches to an answer: 1. there is a complete list from a XVII century author, and 2. a compilation of classical and post-classical names by L&S.
1. Complete list (late Latin)
According to the Harmonia Macrocosmica by Andreas Cellarius (XVII century,) the phases are:
- 🌑 New Moon: Novilunium (Soli coniuncta),
- 🌒 Crescent: Cornicularis (in aspectu sextili),
- 🌓 Quarter: Dimidiata (in aspectu quadrato),
- 🌔 Gibbous: In orbem insinuata (in aspectu trino vel trigono) and
- 🌕 Full: Plenilunium (Soli opposita)
The intermediate ones, in turn, may be crescens (🌓 waxing) and decrescens (🌗 waning; remember Carmina Burana/O Fortuna?) He even goes to name the phase for every single day of the lunar cicle (larger image here.) Note that in English crescent refers to the shape (🌒), while in Latin crescens means growing (i.e. waxing)
2. Older uses:
Novilunium and plenilunium are postclassical, and according to L&S, classical names were luna nova and luna plena.
Luna entry in L&S lists crescens and decrescens, as well as instances of dimidiata (by Cato,) extrema and prima (by Varro) and specific days as counted from the last New Moon (e.g. quinta decima luna, Columella, Res Rustica, 2, 10),