This is a very good question.

I’m sure personal names need no explanation. Chinese monarchs, like other individuals, bore personal names.

Taking your example of Li Shimin, I’ll explain the differences between the different names that Chinese monarchs usually had. First of all, Li Shimin did not reign as “Taizong” because this was a temple name and was accorded to him only after his passing. Therefore, he would not have reigned using a name only to be attained after his death, and he had no way of knowing which exact temple name he would eventually be honoured with.

Temple names (庙号), like posthumous names, were titles accorded to monarchs after their death. It should be noted that temple names were exclusively given to monarchs as well as those who did not rule as monarch in their lifetime but were later posthumously honoured as one. Temple names were originally honoured to monarchs with great accomplishments, hence they were less commonly found in earlier dynasties; however, since the Tang dynasty onwards, honouring late monarchs with temple names became more common. Temple names almost always consisted of two Chinese characters, although in extremely rare cases there were temple names with three characters. The last character would always be either “zu” (祖; meaning “progenitor”) or “zong” (宗; meaning “ancestor”). The former was mostly honoured to the founders of dynasties or those monarchs with extraordinary accomplishments, while the latter was given to the other monarchs. The first character (or first two characters for those rare three-character-long temple names) would be an adjective that describes the reign or character of a monarch. For the case of Li Shimin, his temple name “Taizong” (太宗) is typically translated as “grand ancestor”. Other notable Chinese monarchs that bore the temple name “Taizong” included, but not limited to, Yelü Deguang (of the Liao dynasty), Wanyan Wuqimai (of the Jin dynasty), Ögedei (of the Yuan dynasty), and Hong Taiji (of the Qing dynasty). As a result of Chinese cultural influence, rulers of Korea and Vietnam also assigned temple names to their monarchs. For instance, the temple name “Taizong” is translated as “Taejong” (태종) in Korean and “Thái Tông” in Vietnamese. Since the usage of temple names was often repeated by different dynasties, historians would usually add the name of the dynasty to the temple names for less ambiguity; for instance, Li Shimin is often known as “Tang Taizong” (唐太宗; or “Taizong of Tang”) in order to distinguish him from the other rulers who also shared the same temple name “Taizong”.

Posthumous names (谥号) were less exclusive than temple names, and were honoured to not only monarchs, but also queens or empresses, officials and other noteworthy individuals. The posthumous name bore by Li Shimin was originally “Wen Huangdi” (文皇帝; “huangdi” meaning “emperor”, hence it could also be translated as “Emperor Wen”). Posthumous names were often very elaborate, and could consist of more than ten characters; in such cases, historians would often refer to these monarchs by the last character (or last two characters) of their posthumous names. The words that made up posthumous names were selected to reflect the circumstances of a monarch’s reign. The character “wen” in Li Shimin’s posthumous name is usually translated as “civil”. Monarchs of Japan, Korea and Vietnam also adopted posthumous names. The difference between the purpose of temple names and posthumous names was that the former was the name by which deceased monarchs were known as within the ancestral temple, while the latter was for the purpose reviewing a monarch’s accomplishments (or the lack thereof).

Therefore, both temple names and posthumous names were titles that Chinese monarchs were assigned after their death. There was no way that Li Shimin (and any other monarchs) could have reigned using any one of these titles.

However, Chinese monarchs did have other titles during their reign: era names and regnal names.

Era names (年号) were names adopted for the purpose of year identification. The first to declare an era name was the Emperor Wu of Han, and this practice had since been adopted by Japan (still using this system to this day), Korea and Vietnam. Taking the example of Li Shimin, he used only one era name throughout his reign — “Zhenguan” (貞觀). Monarchs could also choose to change the era name during their reign; for example, Zhao Kuangyin, the founder of the Somg dynasty, used a total of three era names during his reign. Most Chinese era names consisted of two characters, even though era names with three, four, or six characters also existed.

Regnal names (尊号) were titles used by Chinese monarchs during their reign, even though not all Chinese monarchs adopted regnal names. For example, the last emperor of the Liao dynasty, Yelü Yanxi, held the regnal title “Tianzuo Huangdi” (天祚皇帝; or “Emperor Tianzuo”).

As for the courtesy name (字) that you mentioned, this was not exclusively reserved for monarchs. It was common for individuals to adopt a courtesy name upon reaching adulthood. Individuals in Korea and Vietnam also bore courtesy names once they reached adulthood.

For Chinese monarchs prior to the Tang dynasty, it is common for historians to refer to them by their posthumous names. For monarchs from the Tang to the Yuan dynasties, they are commonly referred to by their temple name (with the exception of those who did not have temple names). Monarchs from the Ming and Qing dynasties are known most commonly by their era names. However, it is not wrong to refer to monarchs using any one of their accorded titles, even though the above-mentioned method is the convention currently used by most historians.