The moon being seen during the day is nothing unusual or special. For this reason there is not a special name for the moon on those occasions.
its called a waning
You can see the moon for about 8 hours in the day
You can see the moon sometimes
The only day on which you generally do not see the Moon is on the day of the "new" moon, when the Moon is quite near the Sun in the sky. On any other day of the 29-day cycle, you will - weather permitting! - be able to see the Moon, either in the morning or in the evening.
There's no special name. Stars also come 'out' during the day but they are too dim to overcome the bright blue sky. With binoculars you can see Venus in the day if you know exactly where to look (and don't point the binoculars at the Sun!).
When the moon is visible during the day, it is called a "day moon." This occurrence happens because of the moon's position in its orbit around Earth where it can be seen from Earth in daylight hours.
After a full moon as we see less and less it's called a waning moon
Crescent.
I see the moon during the day often, except on cloudy days. I enjoy looking at the moon. its very pretty and light has nothing to do with it.
Waxing
You can see all three of these. The phase of the moon when it is not visible is called new moon.
You can see the moon in the day if it's in a specific phase and position in relation to the sun. During a full moon, it's visible both day and night. When the moon is a crescent or in a different phase, it may not be visible at night due to its position relative to the Earth and the sun.