In the Chinese lunar calendar, "solar term" refers to a specific moment. The day with the "solar term" is known as the "first day of the lunar month", and the night of the "solar term" is the night of the new moon. "Full moon" also refers to a specific moment. The night of the "full moon" is the night of the full moon. If the "full moon" coincides with daytime, the full moon will occur in the morning; if it occurs in the evening, the full moon will occur in the evening.
Since the orbit of the moon around the earth is elliptical, the time between the solar term and the full moon is not fixed. If the moon is above the orbit during the solar term, it will take longer to reach the full moon because it will cross the apogee, and the full moon may not occur until the 16th or 17th lunar day. If the moon is below the orbit during the solar term, it will take less time to reach the full moon, and the full moon may occur before the 15th lunar day.
In fact, from 2005 to 2020, there were 198 lunar months with solar terms and full moons. The lunar calendar dates of the full moon were: 70 times on the 15th; 95 times on the 16th; 31 times on the 17th; and 2 times on the 14th. Of the 16 Mid-Autumn Festival full moons, 9 occurred on August 16th, 3 on August 17th, and only 4 on August 15th. Of the 16 Lantern Festival celebrations, only 6 had a full moon, with 8 occurring on January 16th and 2 on January 17th. In March 2009 and June 2020, the "full moon" occurred on the 14th. Even if it was the same full moon, it would vary in size due to the different distances between the earth and the moon. The "full moon" when the moon is near the earth, with the positions of the moon, the earth, and the sun close to a straight line and clear weather, is what we see as the largest and roundest full moon. This time may not necessarily be during the Mid-Autumn Festival or the Lantern Festival, and it may also occur in other months.
Therefore, the saying "The moon is especially bright during the Mid-Autumn Festival" may not be true. The phrase "The bright moon rises from the sea, and people on both sides of the world share the same moment" also has its limitations: in different parts of the world, the time when the moon rises from the sea will vary; if the distances are too great, it may be that one place is in daylight while the other is in the night, and it's impossible for both places to see the moon in the sky at the same time.