The same question has been asked in various guises in the past, but never really answered.
We could be forgiven for thinking that there is no satisfactory answer and that the Flat Earth explanation for the phases of the moon is wrong.
Before I start, when I say "for everyone", I do mean for all those over the half of the earth that can see the moon.
We are told
The Phases of the Moon
When one observes the phases of the moon he sees the moon's day and night, a shadow from the sun illuminating half of the spherical moon at any one time.
The lunar phases vary cyclically according to the changing geometry of the Moon and Sun, which are constantly wobbling up and down and exchange altitudes as they rotate around the North Pole.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
When the moon is above the altitude of the sun the moon is fully lit and a Full Moon occurs.
From: The Phases of the Moon but see also The Moon.
In the diagram below I have drawn how I picture the geometry of the earth, sun and moon at the time of a full moon. I have not drawn the moon above the sun, as the time of a full moon the moon would be around 20,000 km from the sun, so a few tens of kilometres up or down could hardly make any difference! If the moon were much higher it would appear much smaller at the time of a full moon - and it certainly does not.
In this diagram horizontal and vertical distances are to scale, but the sizes of the sun, moon and observers are exaggerated, otherwise they would be almost invisible.
So
half the moon is illuminated by the light from the sun. But,
it is illuminated on the side facing the sun and not on the side facing the earth. The observer directly underneath is looking straight up and
can only see half moon illuminated. This observer sees only a
HALF MOON, not a Full Moon.
The other observer, for which the sun would be just setting and the moon rising (or vice versa), sees most of the part of the moon facing him as illuminated, so sees a
nearly full moon.
But, we know for a fact that the
phase of moon does not change (substantially) throughout the night or for observers in different locations, and not as appears here
almost a full moon for those where the moon is near the horizon and only a half moon for those directly under the moon.
Please explain where I am mistaken because this is how I interpret what is said in the Wiki.