What is the difference between Moon sign and Moon phase?
The Moon sign is the Moon's position on the ecliptic, one of 12 signs; the phase is the angular relationship between Sun and Moon, one of 8 stages from new to full. It is possible to be born under a full Moon in any of the 12 signs, they are independent axes.
Why does my Moon sign sometimes come out different between calculators?
Because near the boundary between two signs (the Moon changes every 2 to 3 days), small differences in time zone handling or in the ephemeris used between tools can land on either side of the line; away from the boundary, the result is stable.
Is the Moon the fastest-changing body in a birth chart?
Yes: at about 13° per day, it changes sign every 2 to 3 days, compared to roughly 30 days for the Sun and over a year for Mars onward; that is why the Moon sign needs a more precise date and time than the Sun sign.
Does being born under a full Moon have any real astronomical effect?
The only measured effect is the satellite's visible illumination that night (close to 100%); associations between behavior or personality and Moon phases are symbolic interpretation, not something the astronomical calculation supports.