All sovereigns are denoted by a string of Chinese characters.
1) Han Gao Zu Liu Bang (漢 高祖 劉邦 han4 gao1 zu3 liu3 bang1)
2) Tang Tai Zong Li Shi Min (唐 太宗 李世民 tang2 tai4 zong1 li3 shi4 min2)
3) Wei Wu Di Cao Cao (魏 武帝 曹操 wei4 wu3 di4 cao2 cao1)
4) Hou Han Gao Zu Liu Zhi Yuan (後漢 高祖 劉知遠 hou4 han4 gao1 zu3 liu3 zhi1 yuan3)
5) Han Guang Wu Di Liu Xiu (漢 光武帝 劉秀 han4 guang1 wu3 di4 liu3 xiu4)
The first character(s) are the name of the dynasty or kingdom. e.g. Han, Tang, Wei and Hou Han.
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 Di Cao 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 Zong Li 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.
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.
If you were even more confused after reading above, here is a quick guide (so not a thorough explanation).
1) Emperors before the Tang dynasty: use name of dynasty + posthumous names. eg. Han Wu Di
2) Emperors between Tang dynasty and Ming dynasty: use name of dynasty + temple names eg. Tang tai zong
3) Since all legitimate rulers of China after Qin Shi Huang Di were titled emperor of China, they can also be referred to by "emperor of" and the name of his/her respective dynasty after the temple or posthumous name. eg.
4) Emperors of the Ming and Qing dynasty: use era names (same as reign names) because each emperor has only one distinctive era name. eg. Kangxi (kang1 xi1) Emperor
5) 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.
6) 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. Here is the discussion link.
The tables: the page is pretty long (the second longest wikipedian article). If you are looking for specific monarchs of a dynasty, better use the following "See also" links. The table had been chopped into smaller and digestable pieces according to the dynasties and placed under those pages.