The answer to this is quite simple. China has always had many names and titles to refer to the same particular emperor. For the purposes of Chinese historiography, four are most relevant; the personal name, the posthumous name, the temple name, and the era name. Most, but not all, emperors are most commonly referred to with one of these four names/titles.
- Personal name: Using the personal name of the emperor was largely taboo. Ministers and other members of the court would not be calling their sovereign by his personal name; even later historians would refrain from using the personal name except perhaps in the introduction. This differs from more familiar Western traditions, where monarchs are most commonly referred to with their personal names, nicknames, or some variant thereof. Nevertheless, contemporary historians often refer to certain emperors with personal names, particularly for the founders of dynasties. As you note, Western historians refer to the early Qing emperors who reigned before the fall of Beijing with their Manchu names. This is common for many early steppe leaders, who are often referred to with their native names despite claiming Chinese imperial language. For instance, we talk about Abaoji instead of Emperor Taizu of Liao.
- Posthumous name: Shihao, or the posthumous name, was a title which theoretically reflected the deeds of the monarch and was ascribed to the emperor after his death. Most early sovereigns before the Tang, but not all, are referred to with this: King Zhou of Shang, King Wen of Zhou, Emperor Wu of Han, Emperor Yang of Sui. As time passed, posthumous names became more and more elaborate. While the 2nd century BC Emperor Guangwu of Han's posthumous name consisted of just four characters^1 , the 8th century AD Emperor Xuanzong of Tang had a posthumous name of nine characters^2 , and the 19th century Daoguang Emperor had a posthumous name of 25 characters.^3
- Temple name: Miaohao, the temple name, was also posthumous but much shorter than the shihao: always two characters, the first being an adjective that theoretically described the emperor and the second being a suffix, always either zu meaning "progenitor" (of a new dynasty, or a new imperial lineage within the existing dynasty) or zong meaning "ancestor." There were rules for temple names; most founders of a new dynasty have the temple name Taizu "Grand Progenitor" and the second reigning emperors have the temple name Taizong "Grand Ancestor." All emperors after the Sui and before the Ming are most commonly referred to with temple names; Emperor Taizong of the Tang Dynasty, Emperor Xuanzong of the Tang Dynasty, Emperor Taizu of the Song Dynasty, Emperor Gaozong of the Southern Song Dynasty, so on. Even Ming and Qing emperors are extremely often referred to with temple names, ie you would say Emperor Chengzu instead of the Yongle Emperor. In Korea, Hong Taiji is almost always called Cheong Taejong (Korean pronunciation of Qing Taizong). Certain early emperors, most famously Liu Bang, are also referred to with temple names (Emperor Gaozu of Han).
And the one that answers your question...
- Era name: Chinese emperors issued nianhao, or era names, which was a common way to record dates. An example to help understand how the system worked: the era name zhonghe ("central and harmonious") was introduced in 881 AD, so 881 would be "first year of zhonghe, 882 would be "second year of zhonghe", 883 would be "third year of zhonghe," and so on, until a new era name was eventually issued (as it was in 885). I talk about it a bit here. As I say there, Chinese emperors before the Ming frequently changed era names even under one emperor. This changed with Zhu Yuanzhang/Ming Taizu, the founder of the Ming, who used only one era name, hongwu, for the entirety of his reign of three decades. Hence it made sense to call him the Hongwu Emperor, as he was the emperor of the hongwu era name. Later Ming and Qing emperors kept the tradition of having one era name per reign.^4 As the era name yongle was used throughout the reign of Ming Chengzu we call him the Yongle Emperor, ditto for the era name qianlong and the Qianlong Emperor.
^1 光武皇帝 Guangwu Huangdi. In this context it means something akin to "Restoring and Martial Emperor."
^2 至道大聖大明孝皇帝 Zhidao Dasheng Daming Xiao Huangdi. The meaning is akin to "Way-achieving Great Sagelike Greatly Radiant Filial Emperor."
^3 效天符運立中體正至文聖武智勇仁慈儉勤孝敏寬定成皇帝 Xiaotian Fuyun Lizhong Tizheng Zhiwen Shengwu Zhiyong Renci Jianqin Xiaomin Kuanding Cheng Huangdi. The meaning is something bizarrely long-winded, roughly "Heaven-emulating Divining Centrally upright Rightly-bodied Exceptionally-writing Sagelike Martial Wise Brave Benevolent Frugal Hardworking Filial Clever Generous Succeeding Emperor."
^4 One Ming emperor had two era names, but his reign was an interrupted one; he was captured by Mongols in 1449