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 MoonThe diagram below shows how I picture the geometry(some diagrams in the Wiki might help) of the earth, sun and moon at the time of a full moon. I have not drawn the moon above the sun, as at the time of a full moon the moon would be around 20,000 km from the sun, so a few hundreds of kilometres could hardly make a 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 object (and people) sizes are exaggerated, or else they would be quite invisible.
OK, so you have the half the moon illuminated by the light from the sun. But,
it is illuminated on the side!.
Observers directly underneath are looking straight up and
clearly see only half the side facing them illuminated, so they see only a HALF MOON, not a Full Moon.
The observers, for which the sun would be just setting and the moon rising (or vice versa), see most of the part of the moon facing them 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 countries, 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.