Why Two Moon Phase Calendars Disagree About the Date
Two carefully made moon phase calendars can print the same full moon on different days. There are three reasons, and only one of them is a mistake.
Reason one: the 29.53-day shortcut
The usual shortcut is to count the days since a known new moon and divide by 29.530588. That number is the mean synodic month, the long-run average interval between new moons, and no real interval is exactly that long. The Moon's orbit is an ellipse, so the Moon moves faster when it is closer to Earth and slower when it is further away, and consecutive lunations run anywhere from about 29.27 to 29.83 days.
Within a lunation the error builds up and then unwinds. At its worst the remainder method is roughly fourteen hours from the true time, enough to put a full moon on the wrong calendar day several times a year, and it gives no sign when it has. This one is a mistake, and it is why the moon phase page computes where the Moon and Sun are.
Reason two: time zones
A phase is a single instant. The Moon reaches exactly new at one moment, the same moment everywhere, and only the local date it lands on varies. A new moon at 01:30 UTC on the 12th falls on the 12th in London and Tokyo, and on the 11th everywhere in the Americas. Both dates are right for the same event.
A calendar that converts to local time without saying so gives its readers no way to tell a real disagreement from a time zone one. The moon pages on this site label their times UTC for that reason.
Reason three: what counts as "full"
The four named phases are instants. Many calendars assign the names by illumination band (anything above about 98 per cent gets called full), which spreads a single instant across two or three consecutive days. That is why one calendar can show "full moon" on both the 12th and the 13th while another shows it on only one.
A full moon is also almost never 100 per cent lit. The Moon's orbit is tilted about five degrees to the plane of Earth's orbit, so at most full moons the Moon sits slightly above or below the exact alignment and a sliver of the disc stays dark. A full moon reaches a true 100 per cent only when the alignment is exact, and then it is a lunar eclipse: the Earth's shadow takes the light away again.
How to tell which you are looking at
A calendar that prints an exact time and names its time zone is doing the full calculation. One that prints only a day has either done that calculation and rounded, or used the shortcut, and you cannot tell which. A calendar that names the same phase on two consecutive days is using illumination bands. A calendar that disagrees with a well-sourced one by a day, near a date boundary, in a different time zone, is probably right for its own zone.
The moon phase page here prints the exact instant in UTC and gives a day a principal phase name only when that instant falls inside it. The method page describes the published table its output is checked against.
Equinox and solstice times also differ from one site to another. The equinox and solstice times here explain why: past about 2050 the difference comes from ΔT, the prediction of how fast the Earth will be turning.