You are indeed right in imagining that the styling of the nomenclature for foreign heads of state is a little arbitrary and I hope to illuminate a few points as to why 天皇 was translated as emperor and not king in the eyes of the Europeans.

Firstly,the word 天皇 and the word for emperor in Mandarin ( 皇帝 ) are drawn from the same source,namely the Chinese characters for emperor. It's worth noting that the Japanese had begun calling their emperors 天皇 from a very early period (7th Century onwards),a radical change from the previous title,Yamato-ōkimi,literally,King of Yamato,implying that they viewed their sovereign as equal to the Chinese emperor in stature and majesty.It's thus likely that the Japanese would want foreign delegations to be notified of this desire as well and would communicate that they craved equal status with Chinese rulers to all foreign delegations.

It's impossible to discuss this without mentioning the Portuguese, who made contact with Ming China first in 1514 and translated the Chinese ruler as emperor. When they further encountered the Japanese nearly three decades later,the first Portuguese sailors who reached Japan had to be taught Japanese using Chinese characters . Lidin's book Tanegashima- The arrival of Europe in Japan talks a little about this and,in it,he mentions the first recorded instant of Europeans arriving in Japan.

" On board was a Chinese named Go-hou, who understood writing: from him it was gathered that that this was a nan-banship (‘southern barbarian’ in the Japaneseform of the Chinese words nan-man). On the 26th this vessel was taken to Aku-oki harbour, on the northwest of the island, and Tokitaka,governor of Tanega-zima, instituted a strict investigation concerning her, the Japanese bonze Tsyu-syu-zu acting as interpreter, by means of Chinese characters. On board the nan-ban ship were two commanders,Mura-syukya and Krista-moota; they had firearms, and first made the Japanese acquainted with shooting-arms and the preparation of shooting-powder."

As such,the Japanese had the definitive advantage in dictating the meaning of their language to an otherwise unaware culture.

We now know what the Japanese wanted,but why did the Portuguese agree to this interpretation ? Well,Portuguese Enterprise in the East: Survival in the Years 1707-1757 by Teddy Sim has several ideas on the issue.Drawing from work done by Saldanha,he argues that the Portuguese had a tiered system of the nations they encountered and that this classification was heavily colored by Greco-Roman cultural as well as Catholic religious roots.For example,the Chinese ruler was designated as emperor due to his ruling over multiple ethnic groups and his nation's ties with the ancient Romans. Conversely,the Mughals were never respected as emperor by the Portuguese,with the Mughal emperor's title being 'King of Mogor',never emperor as with the Chinese. They were,however,a step above the other 'petty kings' as they were acknowledged as a step above other rulers,such as the King of Cochin.

In the case of Japan,it'd seem that diplomacy rather than respect or admiration played the greatest role in recognizing the tenno. Teddy argues that the Jesuit perspective on Japan was hopeful,even likely, in the convertibility of its populace to Christianity and were seen to have 'high potential'. In matters regarding East Asia,the Jesuits had great influence with the King and often acted in official and diplomatic capacities for the crown and it was not unheard of for the Portuguese to use cordiality and tactful diplomacy to achieve what it wanted,such as the usage of tencas (a form of tribute) on smaller nations like Cochin.In short,he argues that the acceptance of the term could've been more placatory than anything,as a way to soothe relations.

Finally, the work done by Matsukata Fuyuko and Adam Chulow illuminate on what certain European rulers actually thought of Japan's emperor.If you lack access to it, it's basically a write-up on the implications of an 1844 letter sent by the Dutch king to the Shogun to force Japan to open up its shores or 'fortunate Japan may be destroyed by war'. In said letter,they did not translate the letter on their end and used the Dutch term Uwe Majesteit or 'Your Majesty' not Your Imperial Highness.In the write-up, Masataka also analyzes documents from the Dutch Ministry for the Colonies and notes that,in Dutch circles, King Willem often referred to Japan's ruler(s) as the King(s) of Japan as the Dutch were very much aware that Japan had (technically) 2 rulers split between Kyoto and Edo (the emperor and shogun,respectively)

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I hope this helps!

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Edited :Italicized a few things cuz I'm too used to reddit mobile and discord

Refs

Karl F Friday Japan Emerging: Premodern History to 1850

Olaf G Lidin Tanegashima - The Arrival of Europe in Japan

Teddy Y.H. Sim Portuguese Enterprise in the East: Survival in the Years 1707-1757

Masataka Fuyuko, translated by Adam Clulow, “Reevaluating the “Recommendation to Open the Country”: The King of the Netherlands 1844 letter to the Tokugawa Shogun,” Monumenta Nipponica, 66－1, 2011