Long story short, even though Korea was the only East Asian country directly colonized, the other two were colonized informally, less so Japan than China.
Informal empire is defined as:
a willing and successful attempt by commercial and political elites to control a foreign region, resource, or people. The means of control included the enforcement of extra-territorial privileges and the threat of economic and political sanctions, often coupled with the attempt to keep other would-be imperial powers at bay. For the term "informal empire" to be applicable, we argue, historians have to show that one nation’s elite or government exerted extraterritorial legal control, de facto economic domination, and was able to strongly influence policies in a foreign country critical to the more powerful country’s interests.
(Barton & Bennett 2010)
By the time of Second Imperialism or New Imperialism in the 1800, Empire was changing. Traditional empires like the 13 colonies or the Ottomans were focused on land accumulation and conquest. Land equalled power. Traditional monopolies and agricultural taxes created wealth.
By the 1800s, the model first pioneered in earlier centuries by the Dutch and Portuguese of New Imperialism became more common. This was the idea that the power of a nation was its control over market surplus. Often, through military force, a country could influence another country's trade policy and force them to sell cheaply or buy goods that the government (or people) wouldn't otherwise buy, or be allowed to buy by their government.
For example, Britain was running a trade deficit in the early 1800s with China, as the Chinese produced higher quality products. It started smuggling opium into China, and when the Chinese confiscated British opium, attacked China in the Opium Wars, leading to the opening of Chinese ports and the spread of opium throughout China. Commodore Matthew Perry of the US forced a similar opening on Japan. In Africa, military force was used to extract and dump surplus. The British used proxy rulers, while the French ruled directly through their bureaucracy, both enforcing export monopolies on their colonial zones. This forced the natives to buy their industrial goods. It's no coincidence that imperialism blossomed after the recession of 1873 and a glut in the industrial market: countries needed to find new export markets.
Military control not only let countries keep out their competitors, but helped them extract resources in two ways as well. First, they could build railways and steamship routes to the interior to connect their resources. The Belgians famously built infrastructure only to connect Congo's resources, but not to connect less valuable or poorer areas of the Congo. Second, they could enforce hiring monopolies, or even forced labor on the locals. The most infamous example was the Congo free state. New Imperialism let countries dump their industrial surplus at higher prices, and acquire more surplus at low prices from their colonies.
So did these conditions exist in East Asia?
They definitely existed in China. European powers carved China up into spheres of influence. Sun Yat Sen remarked that China was the most unfortunate colony because it was "everyone's colony" - industry in big cities was regulated by foreigners and illicit goods were essentially legal for the foreigners to sell. During refugee crises caused by flooding and rebellion, droves of Chinese arrived in treaty ports that connected China with other parts of the empire, falling victim to hiring monopolies, and millions became low wage manual workers in other continents or in the treaty ports. China's relative proximity, perception of wealth to steal, and the corruption of its government all played a role in expanding European military dominance in China to be greater than that in Japan.
In Japan, the situation was different. The Tokugawa Shogunate capitulated to Comomdore Perry, but the Emperor (traditionally a ceremonial figure) soon issued the expel the barbarians decree which created an orgy of violence against foreigners. Two southern daimyos revolted, starting the Boshin war that toppled the Shogunate and installed a nationalist government open to intercourse with the West, but constantly and rapidly expanding its own military power. The Europeans did try to stop this, but their influence was limited. The US was preoccupied with the Civil War, while the UK defeated a revolting daimyo in the Battle of Shimonoseki, but did not press its advantage. The Tokugawa Shogunate, backed by France, fumbled the ball so to speak and did not follow the advice of its French military advisors, collapsing rapidly.
As we can see from this comparison, imperialism requires overwhelming military superiority in a local area to manifest, plus the will to use it. In the case of China, full blown military expeditions and landings occurred in the Opium Wars, Boxer Rebellion, and Sino-French war to protect concessions. Even if China wasn't formally colonized, it was informally colonized and Europeans plundered its surplus while dumping their own surplus there. In Japan, even when foreigners were being murdered, the European fleets merely destroyed a daimyo's navy and didn't press their advantage. Once Japan modernized and gained the ability to defend itself, no further concessions could be forced on it.
Sources: (Economic and Social History of the Ottoman Empire, Inalcik)
(Spaces of Global Capitalism, Harvey)
(King Leopold's Ghost, Hochchild)