The king or wang (王 wang2) was the Chinese head of state from the Zhou to Qin dynasties. After that, Wang (sometimes translated "prince") became merely the head of the hierarchy of noble ranks. The title was commonly given to members of the Emperor's family and could be inherited.
From the Han dynasty, huangdi was abbreviated to huang or di. Qing (卿);, Daifu (大夫) and Shi (仕) became synonyms for court officials.
Although formally the son of heaven, the power of the emperor varied between emperors and dynasties, with some emperors being absolute rulers and others being figureheads with actual power lying in the hands of court factions, eunuchs, the bureaucracy or noble families.
The title of emperor was transmitted from father to son. Usually the first born of the queen inherited the office, but this rule was not universal and disputes over succession was the cause of a number of civil wars. Unlike the Emperor of Japan, Chinese political theory allowed for a change in dynasty and an emperor could be replaced by a rebel leader. It was generally not possible for a female to succeed to the throne and in the history of China there has only been one reigning Empress, Wu Zetian of the Tang dynasty.
The first character(s) are the name of the dynasty or kingdom.
e.g. Han, Tang, Wei and Hou Han.
Then come the characters of how the sovereign is commonly called, in most of the times the posthumous names or the temple names.
e.g. Gao Zu, Tai Zong, Wu Di, Guang Wu Di
Then follow the characters of their family and first names.
e.g. Liu Bang, Li Shi Min, Cao Cao, Liu Zhi Yuan and Liu Xiu
In contemporary historical texts, the string including the name of dynasty and temple or posthumous names is sufficient enough as a clear reference to a particular sovereign.
e.g. Han Gao Zu
Note that Wei Wu DiCao Cao never was a sovereign but his son was. Thus he was revered as Wu Di. Cao Cao is good enough for reference.
All sovereigns starting from the Tang Dynasty are contemporarily referred to using the temple names. They also had posthumous names but less-used, except in traditional historical texts. Reversed situation before Tang as posthumous names are contemporarily used.
e.g. The posthumous name of Tang Tai ZongLi Shi Min was Wen Di (文帝 wen2 di4)
If sovereigns since Tang were referenced using posthumous names, they were the last ones of their sovereignties or their reigns were short and unpopular.
e.g. Tang Ai DiLi Zhu (唐哀帝 tang2 ai1 di4李柷 li3 zhu4), also known as Tang Zhao Xuan Di (唐昭宣帝 tang2 zhao1 xuan1 di4), was last emperor of the Tang Dynasty reigning from 904 to 907.
Han Guang Wu Di is equivalent to Dong Han Guang Wu Di since he was the founder of the Eastern Han Dynasty. All dong(east)-xi(west), nan(south)-bei(north), qian(former)-hou(later) conventions were invented only by past or present historiographers for denoting a new era of a dynasty. Never used during that era.
Emperors of the Qing dynasty: use era names (same as reign names) because each emperor has only one distinctive era name. eg. Kangxi (kang1 xi1) Emperor
Overrides rules 1 to 3: If there is a more common convention than using posthumous, temple or era names, then use it. eg. Cao Cao instead of Wei wu di.
Some scholars prefer using the Wade-Giles romanization instead of the Pinyin but the above formats still hold. eg. Han Wu Di = Wu-ti Emperor of Han Dynasty.
If you prefer better clarification and do not want to bother with all those "names", please refer to each dynasty of which the specific convention is shown on top of its sovereigns. Hopefully it helps you.
However a consensus has yet to be reached in Wikipedia. .
