How was the term "Mandate of Heaven" used during the rise and fall of Chinese dynasties?

Question

The Mandate of Heaven (Chinese: 天命; pinyin: Tiānmìng; Wade–Giles: T'ien-ming; lit. 'Heaven's will') is a Chinese political philosophy that was used in ancient and imperial China to justify the rule of the King or Emperor of China.
_{Mandate of Heaven, Wikipedia, sourced 7 December 2021.}

A Reddit post recently flatly contradicted the above claims about the "Mandate of Heaven":

Most western history books and videos use the term "Mandate of Heaven" to describe Chinese Dynasty cycles. However, Strangely most Chinese Students and Scholars have never heard this term before, as the term "mandate of heaven" was never used in actual historic context. ...

In Short, Tianming was rarely used in any history context to describe anything related to rise and fall of Chinese dynasty.
_{Fake Translation Debunked: Mandate of Heaven by user u/MingoUSA at r/ChineseLanguage, c. 7 December 2021.}

It's hard to tell precisely if this claim is correct. The Wikipedia page doesn't describe the origin of the term "Mandate of Heaven". So...

Question: How was the term "Mandate of Heaven" used during the rise and fall of Chinese dynasties?

Answer 1

A mandate is something that is granted but can be revoked. The Mandate of Heaven implies a sort of reciprocity. The people must obey their Emperor, but the Emperor must also be just or he may loose his "mandate". "The continuation of the mandate was believed to be conditioned by the personal behaviour of the ruler, who was expected to possess yi (“righteousness”) and ren (“benevolence”). If the emperor’s personal life became immoral or his rule tyrannical, Confucianists taught, he had not only lost his right to rule but should be removed by revolution, if necessary." (1)

According to professor of Chinese intellectual history at the University of British Columbia Josephine Chiu-Duke Official Confucianism, which emerged under the strong-willed Emperor Wu Ti in the second and first centuries BCE, claimed to be based on Confucian teaching. However it “absolutized” the relationship between the emperor and subjects, rulers and minister, fathers and sons, and husbands and wives. “The minister is bound by the ruler and so on and so forth,” she said. ”In other words the reciprocal relationship between ministers and rulers, sons and fathers, and wife and husband no longer exists.” (2)

I would guess that each time a dynasty was overthrown the new Emperor would refer to the old Emperor having lost his "mandate". Later he would downplay the "mandate" part and instead refer to himself as Son of Heaven.

(1) Tianming

(2) UBC scholar explains how official Confucianism diverges from classical Confucianism—and why this matters in China