Thousands.
See related link for a picture.
Please update this answer with the amount you have counted.
The side of the moon that faces Earth (the near side) has more craters compared to the far side. This is because the near side is more exposed to impacts from space debris due to its proximity to Earth, while the far side is shielded from some impacts by the moon itself.
The far side of the Moon has more craters because it lacks the large expanses of dark, smooth volcanic maria that are present on the near side. These maria shielded the near side from impacts, preserving its surface. Additionally, the far side is more exposed to incoming asteroids and meteoroids due to its position away from Earth, leading to more frequent impacts.
There can be craters anywhere on the moon, not just the side we see
The moon has a geosynchronous rotation as it orbits the earth, meaning the same side of the moon always faces the earth. The side we don't ever see from earth is known as the "far" side of the moon. The far side has more craters because it is not protected from debris by the earth.
Almost all the maria are on the near side. The far side is mostly just craters.
Both sides of the moon are pocked with craters, but the far side tends to have more large, basin-sized craters. These large craters can provide shielding from disruptive signals from Earth, making it an ideal location for telescope installations aimed at observing deep space.
The near side of the Moon is the one we see; maria (seas), highlands, craters, and mountains. The far side is much different. The far side is more mountainous, with fewer obvious craters and very few plains or flat areas.
The far side of the moon has a higher concentration of impact craters compared to the near side. It also has fewer maria, or dark flat plains formed by ancient volcanic eruptions, making it appear more rugged and heavily cratered. Additionally, the far side lacks the large basins like Mare Orientale that are found on the near side.
The side of the moon that faces Earth is smoother because it has been bombarded by meteorites and lava flows, which filled in many of the craters. The far side of the moon, on the other hand, is more rugged because it has been shielded from Earth's tidal forces, resulting in more craters and fewer lava flows.
The Moon's crust on the near side is thinner and smoother with large plains called maria, formed from ancient lava flows. The far side has a thicker crust with more craters and mountains, as it has been shielded from volcanic activity due to tidal locking.
There are two separate things that need to be addressed. One is the surface area of the Moon. The other is the surface area of the NEAR SIDE of the Moon. As a sphere in space, the Moon is always 50% illuminated, except during lunar eclipses. The phases of the Moon measure what percentage of the NEAR SIDE of the Moon is illuminated. At the new moon, none of the near side is lit up; at the full moon, 100% of the near side is illuminated.
The Moon's crust may be thicker on the far side. So there was less volcanic activity. So there is less lava to cover up craters. Lava flows form the "maria" which are much more in evidence on the side facing Earth.