What do you call this oriental building in English?
Gazebo or pavilion doesn't go with this building, I think...
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My answer is based on what a Westerner would call such a building (common usage), not on whether the origins of the word are factually correct.
Somewhat analogous to how "citizen" technically refers to the inhabitant of a city, but we commonly understand that it can refer to the inhabitant of a nation as well.
Or, a bit more thematically linked to your example, why we call the mountain in Japan "Mount Ōyama", when "Ōyama" already inherently means mountain in Japanese. It's a westernization, it forgoes factual correctness in favor of using terminology that feels more correct to the ears of a Westerner.
My first instinct would be to call it a pagoda.
A pagoda is a tiered tower with multiple eaves, built in traditions originating as stupa in historic South Asia and further developed in East Asia or with respect to those traditions, common to Nepal, India, China, Japan, Korea, Vietnam, Burma, Sri Lanka and other parts of Asia.
However, please do consider that this answer is from a Westerner's point of view. I'm not sure if a pagoda is an overgeneralization.
Similarly, Wikipedia lists a pagoda as a building with multiple tiers of eaves. I'm aware that the building in your picture only has one tier, but I doubt that anyone in the West would refrain from calling it a pagoda because it only has one tier.
Also, I'm not sure if you are focusing on the specific architecture of the picture you linked. As you can see on the Wikipedia page that I linked, pagodas come in different architectural styles.
From your suggestion, I do think that pavilion is a good option too.
3a : a usually open sometimes ornamental structure in a garden, park, or place of recreation that is used for entertainment or shelter
3b : a temporary structure erected at an exposition by an individual exhibitor
e.g. visited the Chinese pavilion at the international exposition.
It's interesting that the example itself refers to a Chinese pavilion. Although the building in your picture looks more Japanese than Chinese, the Google Images results for Chinese pavilion are very close to it:
I would favor calling it a pagoda, as this name makes an implication about its Oriental architectural style.
Doing some research on it now (because of your question), it seems that pagodas are inherently towers, and I would not describe the building in your picture as a tower.
Nonetheless, I do think that pagoda still applies, because I suspect that it will be interpreted by an English speaker to mean "an ornamental building in an oriental style", without necessarily expecting it to be a tower.
Pavilion, while correct, would be less understood by an English speaker (as they will primarily think you are talking about western pavilions). However, if you describe it as a Japanese pavilion, it seems almost equivalent to what most people think of as a pagoda.
Normal practice would be to describe the building with the translation of the description in the native language. That would describe the building's purpose.
There are no specific English words to describe specific types of architecture other than loan words from the relevant language.
That building looks familiar to me as one I have visited in Kyongju. My understanding is that such buildings are meeting halls or ceremonial locations. They would be described in that manner.