In most scholarly works, there will be a section on definitions and the choices involved in defining as such. This is pretty standard practice for historians because there is tremendous ambiguity between the terms in English and the terms in the original source language.

For example, in my own research, I define an emperor in the Late Roman period as someone who is mentioned in the sources as an imperator/autokrator, a caesar/kaisar, or an augustus/augoustos/sebastos.

I specify Late Roman because in the early imperial period, an imperator could just be a military commander honored with that title after a successful victory. Germanicus was hailed an imperator although he wasn't what we call an "emperor." That would be Tiberius, who followed Augustus' convention of considering himself a princeps.

So your question is a correct one to ask. The answer is simply: every historian has to decide for him/herself, and ideally should define their terms and their choices for usage in the published work.

A partial answer, albeit with respect to societies with more close contact to European society, can be found in this post.

https://www.reddit.com/r/AskHistorians/comments/18qygi/imperial_titles/

This post from a few months ago by /u/Opechan should answer part of your question. They explain that the use of "chief" is a highly politicized term in recent historiography, and in fact the early English settlers of America typically did refer to Native leaders as kings and emperors at first.

As for the other half of your question, I think the difference between a king and an emperor is often purely semantic, and when talking about historical societies, you usually just have to go by that society's nomenclature. However, if a distinction can be made, I would say this:

A king is any ruler with absolute or almost absolute executive power. I say almost, because a constitutional monarchy is still a monarchy. Kingships are usually dynastic, but sometimes kings are also elected.

An emperor generally has the added quality of ruling over an extended empire, while a king's domain can be as small as a single city. Empire entails the sovereignty of one society over other societies, varying in language and culture, that formerly had sovereignty over themselves. The emperor may allow these conquered societies to retain their political systems (even their kings) and simply pay tribute, or he may install representatives of the central government, such as prefects.

EDIT: I would also add that some societies have rulers that may as well be kings, but refuse to call themselves that for political reasons. Augustus, the first emperor, called himself "Imperator," which originally meant "commander," specifically to avoid the connotations of the word "Rex," which left a bad taste in Roman mouths due to their republican history. Nowadays, dictators invent titles for themselves to preserve the facade of democracy, but Kim Jong-Un is basically the king of North Korea, by any reasonable definition.

Native American studies major here. So for many Amerindian tribes, the word "chief" and such were not an exact match for such English terms. The Aztec "chief" or "emperor" as they are mostly referred to, were called "Tlatoani". It basically means "the one who has the word" or more loosely "speaker". Moctezuma II and Cuauhtemoc were the two most notable Tlatoanis that history had remembered.

For other tribes the similarity of terms of important tribal leaders is a bit closer to English. For the Lakota tribe, they call their "medicine men": Wicasa Wakan (literally "Sacred Man").

I apologize for no exact sources for this. I am on mobile right now. I spent considerable time with both tribes as well as my field of study.

For China, Huangdi (The Chinese word commonly translated as Emperor) was an official title, invented by the ruler of the Qin dynasty. To understand the title of Emperor, you'd also have to understand the concept of the Mandate of Heaven. The idea was that Heaven itself bestowed rulership upon the rulers, and if the ruler abused it's power or was otherwise unqualified, Heaven could take it away and give it to another man who'd prove himself worthy. Signs that the Mandate of Heaven was lost included disasters, riots, famine, war, and omens. If you sat on the throne, and the land was in peace, then obviously you were the designated emperor. If you sat on the throne, but corruption was rampant, you can't protect your citizens from foreign invaders, dams are breaking, and there is an ill-forboding meteor seen in the sky, then your citizens would believe that you lost the mandate of heaven, and thus the title of emperor. The next person to sit the thone of a united and peaceful China could then claim to be the next emperor. Thus, the foreign conquerer Kublai Kahn, being able to sit the throne in a united China, eventually laid claim to the title of emperor.

United is a keyword here, since during large swathes of Chinese history, China concisted out of groups of kingdoms. The rulers could claim to be emperors, but often are referred to as kings instead. As a rule of thumb, the ruler of a largely united China, laying claim to the title of Emperor, is called an Emperor. The rulers of fragmented parts of China are usually called Kings. If there are large fragments (like one third or half of China under one ruler), the rulers of which have laid claim to the title of emperor, and they rule for a longer period of time (e.g. Three Kingdoms, North and South Song dynasty) they might also be called emperors.

*Edit: like /r/networkllama pointed out, the literal word "Emperor" was not invented by the Chinese, so I cleared that up in line one.

To be a bit more specific about China, the standard work of reference is Charles O. Hucker's A Dictionary of Official Titles in Imperial China. This covers thousands of years of titles and assigns consistent English translations to a huge number of positions in the imperial hierarchy. In total, there are 8291 entries.

Below is his entry for Huángdì 皇帝 (Emperor). Note: Hucker used the now-outdated Wade-Giles romanisation system of Mandarin Chinese rather than pinyin, so his dictionary romanises this word as huang-ti. The pronunciation is the same.

Lit., a combination of terms designating legendary sage kings of highest antiquity: Emperor, from Ch'in on the standard official designation of a dynastic ruler; instituted by the First Emperor of Ch'in to supersede the earlier term King (wang).

In pinyin, Ch'in is Qín. Here's his entry for prefect:

chīh-fŭ 知府 SUNG~CH'ING: Prefect of a Superior Prefecture (fu) from Sung through Yuan, of any Prefecture (also fu) in Ming and Ch'ing; a special duty assignment for a high-ranking official with a nominal post elsewhere in the governmental hierarchy in Sung; a regular post thereafter, rank 4a in Yiian, variable from 3b to 4b in early Ming, then 4a till 1753, then 4b. P32, 49, 53,72.

So why "prefect"? Well, we already know the meaning of fǔ (an administrative division, translated as Superior Prefecture or Prefecture, depending on era). Hucker explains the word zhī 知 like this:

Lit., to know, to take notice of: from Han times on commonly used as a prefix to an agency name in the sense to manage or to administer, often in the form chih ... shih (managing the affairs of ...), i.e., Administrator of .... Originally suggested a specially authorized appointment of someone with nominally different status to serve in a normally more prestigious post as administrator of an agency: e.g., shih-chung chih tung-kung shih (Palace Attendant and Administrator of the Eastern Palace) in Han, chieh-tu fu- shih chih chieh-tu shih (Vice Military Commissioner and Administrator of the Military Commission) in T'ang, chi- shih-chung chih chien-yŭan (Supervising Secretary and Administrator of the Remonstrance Bureau) in Sung. Not later than Sung times, some titles of this sort became regularized: e.g., chih-hsien (District Magistrate). Although the term chih normally designated the official principally in charge of an agency, such was not always the case; e.g., see chih-kuan (Vice Superintendent) of Ch*ing times. P5, 6, 12, 15

So you can see that the problem is that similar terms get used in slightly different ways in different eras. The same word might be translated one way in the context of one title, and another way in a different context. Because Hucker translates fǔ as "prefecture", it makes sense to simplify "administrator of a prefecture" (zhīfǔ) as "prefect".

This isn't to say that Hucker's work is without flaws. In a review, Edwin Pulleybank summed it up by saying:

As he states in the preface, in his choice of English translations the author has tried to steer a middle course between etymological exactness in rendering the Chinese words and making misleading equations with superficially similar, but essentially different terms borrowed from familiar Western political terminology. He prefers renderings that suggest the function of an office as opposed to ones that provide literal equivalences for the Chinese words. This will not please everyone. Indeed, his choices may not please anyone all the time.

Despite some of the issues and disagreements, Hucker is usually the first port of call for a historian of imperial China working in English, and the greatest use of the book as standard is the consistency among translations.

Sources:

• Hucker, Charles O. A Dictionary of Official Titles in Imperial China, Stanford University Press, 1985
• Pulleyblank, Edwin G. 'A Dictionary of Official Titles in Imperial China (review)' Pacific Affairs, Vol. 60, No. 2 (Summer, 1987), pp. 325-327

African titles can be really hard sometimes. They're context sensitive, which did not make them very transparent to colonial regimes or ethnographers, so often they became "chiefs," "paramount chiefs," or "kings" depending on what was administratively needed; alternately, the decision could come from missionaries who chose to define translations. So, for example, if you're talking about Venda, you have khosi, which is called a chief. However, the effective spiritual ruler (and sometimes direct ruler, via familial and client connections) was also just a khosi. So the creation of a term--khosikulu or khosikhulu (see what I did there?)--was necessary to define such a position even though it was never used as an official title. What's worse is that a great khosi might not always have a title that would retain such greatness, so its position as primus inter pares (if I can go Latin for a second) could not be presumed. The Venda title for a king, Thovele, is also that used for God in the Christian sense, a single ruler over all. It is not quite historical, in that it didn't exist for a very long time and was never without dispute at the time it existed even if now, looking back at the great king Thohoyandou, it gets used. Besides mahosi (the plural of khosi) you have magota (singular gota), who are powerful local leaders and lineage heads. If a khosi is just a chief, what are the magota? Headmen? They're recast that way under colonial rule, but that's not quite the original stance, and a local gota could be extremely powerful. This is before we even touch the question of women's ranks--from makhadzi and ndumi (sister of the khosi, daughter of the khosi) on downward--which colonial ethnography converted largely into ceremonial familial lines and ignored for their political capacity to determine the policy of an entire kingdom. Then there's a social-class aspect to consider, de facto vs. de jure power, you get the idea.

Effectively, historians then must make decisions about what term to use for these. One South African who's written a great deal on Venda chooses to retain the Venda terms while recognizing their flexibility. Venda historian-ethnographers (Mpaya Nemudzivhadi and Victor Ralushai, in this case) flip back and forth in use and prefer to describe the particulars of an individual, or hold very detailed discussions of meaning, respectively. Still other authors follow the Paul Landau model and say "look, there's no way we can use the African terms as shorthand if we're going to cover all the variability here. Let's just use the English and make sure we note that it's not exactly what we normally think of, and explain those terms in the process, so as to keep the barrier to entry low for readers." The baggage of the colonial era is so incredibly heavy that it's changed the meanings of many of the titles over time, dictated their usage, and by association it's dictated how we have to try to respond to the erasure of meaning through those things. If this seems confused, inconsistent, and impossible to do in a way that will please everyone, then you understand.