The thickness of the crescent depends on the age of the new moon at the time of observation. For the specific location where you are, the young crescent moon has the least age in some months and a greater age in other months. In some months the age might be just below the visibility range (about 12 hours) and you will not see the crescent. Next day, the age will have increased by 24 hours, to reach 36 hours, and the crescent will be easily visible if the sky is clear. The 36 hour moon will look so big that many people will call it the second day moon, but, for your specific locality, it is still the first day moon. In some localities, the moon might have attained 17 hours of age on the previous day and might have been seen as a thin crescent. The second day moon in these localities will have 41 hours of age. So you see, depending on where you happen to be, a 36 hour old moon may be a first day moon and a 41 hour old moon, which is not much different from 36 hours, might be a second day moon.