I can't answer your full question, but I did want to make one note about your premise and one of its unspoken conceits. Japanese culture is seen, especially in the West, as something of a monolith ethnically. Reasons for this vary, due to both an ignorance of subsets of Japanese people and also, for example, the noted extreme resistance to letting outsiders fully integrate into society -- such that ethnic Koreans who have lived in the country for whole generations are not considered fully Japanese (look up Zainichi).

So just who are the Japanese people, then? Taken from the introduction of Ruins of Identity: Ethnogenesis in the Japanese Islands:

Sasaki Komei, one of Japan's leading anthropologists, defines the Japanese as those people "who speak the Japanese language as their mother tongue, who possess traditional Japanese culture, and who think of themselves to be Japanese" (Sasaki 1991b, 12). Though Sasaki admits this is a "loose definition" [...] minorities in Japan speak Japanese as their mother tongue, yet few would probably see themselves as embracing "traditional" Japanese culture, however that is not defined. Many of these minorities may think of themselves as Japanese citizens, yet they are essentially views as foreigner by most "ethnic" Japanese.

and

Kita Sadakichi, who defined Yamato minzoku (the Japanese race) as a "general term for the whole populace which has lived in our island country for many years, spoken the same national langauge (kokugo), possessed the same customs, considered themselves as a single people, and also reverently accepted the unbroken imperial lineage of the emperor" (Kita 1978, [1929]: 211).

I bring this up because your question seems to be founded on the idea that the Emperor ruled only the Japanese people, where "Japanese" is probably being used herein as interchangeable with the Yamato ethnic group who, as we see above, are defined in a nebulous sense that is the product of years of evolution. While this ethnic group is incredibly dominant in modern Japan as a percentage (well over 90%+), this fails to account for Ainu, Ryukyuans, Nivkh, and Orok peoples along with the better known populations of Han Chinese, Koreans, etc.

One issue is that what makes one an "emperor" versus a "king" in Japanese-translated-to-English terminology may very well be almost entirely a choice in the flavor of transliteration, by historians writing in English, that was set to demarcate the creation of a more grand or pompous monarchy than that which existed prior to Emperor Jimmu. If we cruise past that obstacle, then it would appear that one could argue that by ruling over disparate ethnic groups or different cultures even within what would eventually become the conglomerated Yamato people of today then Jimmu's court certainly satisfied the requirement of ruling peoples outside of its regional home base. After all, one of the concurrent background events during the time of Jimmu's conquests is the emigration of the Yayoi peoples to Japan, mingling with the Jōmon people to eventually become the modern Japanese.

Getting further into what makes the Japanese Tennō translated to Emperor (rather than king) would require me to start postulating outside of what I can readily determine, such as that this is to emphasize the idea of his heavenly appointment or having a religious mandate to rule rather than rule by the right of earthly might.

I hope this at least somewhat satisfies you, or instills you with the desire to research more.

Sources I used:

RUINS OF IDENTITY: ETHNOGENESIS IN THE JAPANESE ISLANDS by Mark J. Hudson ISBN-13: 9780824821562 Published: August 1999

A History of Japan: From Stone Age to Superpower. Kenneth Henshall London: Palgrave Macmillan. ISBN: 978-0-230-34662-8. Published: 2012

edit: changed the wording on some of this for clarity.

I would note that title translation plays a role here, and that historically, other countries did use the equivalent title to King rather than Emperor in discussing Japan.

This goes into more traditional Chinese worldview and the sinocentric model, but Japan was historically called the Kingdom of Wa in Chinese prior to the Tang dynasty and its emperors and empresses were indeed referred to as Kings and Queens in Chinese (倭王, the character 王 meaning King).

One thing to keep in mind is that translation often requires one to make choices, and that many titles don't translate equally. If you go back to the era of Medieval Europe, there was only 1 emperor and that was the Roman emperor. This tradition lasted a long time; it isn't until Napoleon destroys the HRE do we see massive titular inflation as within a century, we see the French Empire, the Austro-Hungarian Empire, and German Empire all form. In contrast, Denmark, Spain and Portugal were most definitely empires at the heyday of their colonial exploits yet never formally took such a term. There is the definition today of what an empire/emperor is in "theory," but it doesn't really match with how it was used historically.

The Japanese-language title most often associated with the emperor, tennō, actually implies heavenly rulership, not territorial hegemony. This is a concept most likely drawn from Chinese political and philosophical texts known at the court of the Yamato rulers (the first Japanese monarchs to adopt the imperial style) in the 600s CE. During this period, the Sui and Tang dynasties in China exported political and cultural institutions throughout their zone of cultural influence, including to Japan.

Initially, one of the main Chinese imperial titles, tianzi, or Son of Heaven, (tenshi in Japanese) was utilized by the Yamato rulers to establish parity when conducting diplomacy with their Chinese counterparts. There is a famous exchange between the Yamato empress Suiko and the Sui emperor Yangdi in which she identifies herself as the tenshi/tianzi of the "Land of the Rising Sun" addressing the tianzi/tenshi of the "Land of the Setting Sun." See Joshua Fogel, Articulating the Sinosphere, for more on this.

Later in the 600s, Yamato rulers adopted the tennō title (also the title used today, after an intermediate period of disuse, and which implies "heavenly ruler"). In his Imperial Politics and Symbolics in Ancient Japan: The Tenmu Dynasty, 650-800, the scholar Herman Ooms suggests three potential origins (each in Chinese philosophical, religious, and political practice) for this particular Japanese title, which aligned with the Chinese title tianhuang.

1. The title tianhuang described a divine figure likened to the Chinese emperor in a historical text that would have been known in Japan.
2. The title tianhuang was used in Daoism in association with beings of elevated spiritual status. Ooms identifies a potential link between related Daoist practices and the Yamato court.
3. The title tianhuang was adopted as a title by Emperor Gaozong of Tang, who reigned during a period in which this title was used by the contemporary Yamato ruler, Emperor Tenmu.

Ooms provides evidence suggesting all three hypotheses but thinks that the first was most likely (p. 155).

what kind of meaning did the Emperor's title have in the Japanese language

To elaborate on this question, in Japanese there are two different words, 天皇/Tennō, referring to specifically the Emperor of Japan, and 皇帝/Koutei, referring to what we would typically consider an Emperor in say Rome. So in Japanese these are not the same thing, and it's purely a translation choice that these refer to the same word Emperor in English. The term Mikado (帝) was once used in English around the 19th century to refer to the Emperor of Japan, but is no longer common.

While 皇帝/Koutei is purely a political title, to refer to a "King of Kings", 天皇/Tennō has both political and religious connotations - the 天/Ten character literally means "Heaven". Thus, the title can be translated as "God-Emperor". This has its basis in the oldest myths of traditional Japanese mythology, as depicted in the Kojiki and the Nihon Shoki, which can be thought of as the "Japanese Bible" in how they form the basis for the religion. Within these works, it is stated the founder of the Imperial Line, Emperor Jimmu, was a direct descendant of the Great God Amaterasu, and thus the Japanese Emperors have a divine right to rule.

In this sense, it might make more sense to compare them to the Pope of the Catholic Church rather than western Emperors, they bare many striking similarities in claiming divine right as conferred by their respective religious deities, and in having a more spiritual and symbolic position rather than the direct political leadership of western emperors. In many periods of Japanese history the Emperor has been seen as little more than a figurehead, and the current shōgun or military government had effectively complete power over the country - even after the Meiji Restoration, which was intended to restore imperial rule, it is debated by historians just how much practical power the Emperors wielded compared to the government of the time.

In short, no Japanese would consider the 天皇/Tennō and a 皇帝/Koutei to be the same thing, and it's the translation where the distinction becomes blurred.