How to map moon phase number + "lunation" to moon phase name?
The WeatherBit 16-day forecast documentation gives these output field descriptions:


moon_phase: Moon phase illumination fraction (0-1)
moon_phase_lunation: Moon lunation fraction (0 = New moon, 0.50 = Full Moon, 0.75 = Last quarter moon)


The illuminated fraction doesn't help you distinguish waxing and waning phases, so moon_phase_lunation is the value to use.
One possible mapping is:
0.0  0.06  0.19  0.31  0.44  0.56  0.69  0.81  0.94  1.0