Yes and no. The educated class in Korea did see themselves as "inferior" to the Chinese in terms of cultural advancement and attainment of Neo-Confucian knowledge - they were, after all, not China, but they knew the Chinese did not see them as "uncivilized."

The Joseon had their own hierarchy of who was civilized and who was not based on how in tuned they were with Chinese culture, and they ranked themselves as number two, second only to China. This was well documented in Korean sources:

From the Dongmongseonseup (a Neo-Confucian textbook for children):

我國雖僻在海隅，壤地褊小，禮樂法度，衣冠文物，悉遵華制。人倫明於上，教化行於下，風俗之美，侔擬中華，華人稱之曰小中華。

Translation: Even though our country is secluded along the coast, our lands [territory] narrow and small, our rituals and music, laws and institutions, clothing and literature, all follow the Chinese model. Enlightenment spreading up to the upper classes and education spreading down to the lower classes, such beautiful custom is similar to China, so the Chinese refer to us as 'Little China.'

From the Joseon Sillok, Seongjong Sillok:

吾東方自箕子以來，教化大行，男有烈士之風，女有貞正之俗，史稱小中華。

Translation: We came from Jizi [legendary figure from the Shang Dynasty] in the east, are educated in important affairs, our men are customarily courageous and our women are chaste and upright in their behavior, and thus, history [or historians] refers to us as 'Little China.'

From the Joseon Sillok, Jungjong Sillok:

我國家邈處日域，自中國視之，與安南、交趾無異，而中國之人不鄙夷之，至稱小中華。

Translation: Our country is remotely located by the border of the sun [remotely located in the east where the sun comes up, but could also mean located by Japan]. From the Chinese point of view, we are no different from Annan and Jiaozhi [modern day Vietnam]. Yet the Chinese does not look down upon us as barbarians and instead refer to us as 'Little China.'

In fact, the Koreans even considered themselves to be the true successor to the Ming after the Manchus took over China.

For Japan, it was much more complicated. Japan was never really ingrained into the Chinese world order and so both the Chinese and the Koreans saw them as uncivilized barbarians. The Japanese were aware of this, but it did not seem like they really cared. The Tokugawa actively used Chinese Neo-Confucianism to administer their state and many Tokugawa scholars supported the study of Chinese classics, philosophy, and literature, but since the Tokugawa regime never bowed to the Chinese, they were still seen as barbarians.