Struggling to understand the phases of the moon Source

Typically this diagram is how the phases of the moon are explained but the issue is this, namely, the phases seen on the right half are not possible. All the phases should appear on the left half during the night when the moon is typically most visible. 

So how on earth are the phases even generated? 

The most remarkable phases are waning moon phases because the moon is lit up on the face away from the sun light source. 

What am I missing? 

I’ve edited the question by adding two additional photos showing the progression of the moon in the night sky over the course of a given time interval and several days, as well as a synodic month which would hopefully indicate based on the position of the moon the sequence of days and based on the height the time of day/night. 


Source


Source

Here is a link to the explanation of how she generated her photo and over what time span but you must have a Facebook account to read it. It is easier to follow the Italian than the English but both are available

Explanation
As noted in a comment, you should imagine the picture is looking down on the Earth, and remember that the actual moon is much further from the Earth, relative to their sizes.

Imagine a person standing on the left side of the Earth (in the diagram). The time would be midnight for them. They could see the moon when it is full, and they could see the moon when it is a gibbous moon (but the gibbous moon would be lower in the sky)  They could even see the moon at first quarter, but it would be on the horizon.  You can't see a crescent moon at midnight.

Now imagine a person standing in the top left on the Earth (in the diagram). The time for them would be about 9pm. (and night time) That person could see the half moon and even a crescent moon (but the crescent would be on the horizon)

It is only possible to see a crescent moon at dawn or dusk, you never see a crescent moon at midnight. At 6pm (at dusk) you can see the crescent moon. If you look to the moon, you see mostly the side in shadow, but a little of the light shining around the side, so it looks like a crescent.

Now you are also confused about the illumination side shown in the waning phases, This is because the picture shows the view of the moon that a person would see if they were standing on the Earth. Because that person is upside down (from our point of view) they see the moon reversed. In other words, from our position above the North Pole, we see the right side of the moon illuminated.  From their position, it is the left.

If the diagram still confuses, discard it and think in terms of angles.

From the perspective of the Earth, the sun and moon go around in circles at different rates, so the angle made between the sun and the moon changes. When the angle between the sun and the moon is about 90 degrees, then half of the near side of the moon is lit by then sun, and a half moon is visible.

When the angle is close to 180 degrees, then nearly all the moon is lit by the sun, and the moon is full.

When the angle is close to 0 degrees, then none of the near side will be lit, and the moon won't be visible.

When the angle is small, ie the moon is close to the sun, then a crescent moon will be visible. A crescent moon can only be seen if it is near to the sun, so the sun cannot be far below the horizon. This means that the a crescent moon cannot be seen late at night when the sun is far below the horizon, but at some times and locations the sun might not set until late in the evening, so the setting time of the moon might be quite late.

Nevertheless it is the angle between the sun and the moon that determines the phase of the moon, and nothing else.