aCalendar displays the moon phase in week view and month view (configurable in settings - design - week & month).
Moon phases are computed for your device timezone (the error should be less than 2 minutes, so the date should be correct 99.9%).
If you do find a difference to some other calendar, please trust aCalendar or check with NASA or TimeAndDate as reference.
aCalendar uses the common astronomical symbols, where the full moon is displayed in light color and the new moon is displayed in dark color (corresponding to full and new moon being light and dark). The exact colors may depend on the color theme selected.
Full moon: ⚪ dark theme ⚪
New moon: ⚫ dark theme ◯