The usage of the characters 文化 as the translation of the modern, Western concept of culture does come from Japanese usage. Although the word 文化 existed in Chinese, its meaning was related but not exactly corresponding to the modern definition.
"Culture" is not necessarily a "central" word to a culture. In fact, the usage of culture in its modern sense (instead of the sense in e.g. agriculture) only took on in the Western world around 18th century after Enlightenment.
The modern Western sense is an extended metaphor based on the idea of the cultivation of mind. In a poetic parallel, this idea is quite close to the some usages of 文化 in Chinese:
We look at the ornamental figures of the sky, and thereby ascertain the changes of the seasons. We look at the ornamental observances (文) of society, and understand how the processes of transformation (化) are accomplished all under heaven. I Ching: Bi (1000–750 BC) [Translation by James Legge]
When the sage governs the world, he prioritizes moral virtue over military force. Using military power is for those who are non-compliant. If the transformation (化) using moral virtues (文) fails, punishment shall then be imposed. Garden of persuasions: Military affairs (The author lived 79-8 or 77-6 BC)
The idea of transformation (化) using moral virtues (文) can be compared to the cultivation of the mind.
However, at the times of these writings, there was no distinct concept of "culture" in its modern sense in China (nor in Europe, for that matter).
In the 19th century, China and Japan both experienced significant transformations. The interactions and exchanges with Western nations intensified, especially in Japan.
Introducing Western ideas to the East, both Chinese and Japanese scholars created or repurposed existing words. Since Japan was at the forefront of Western contacts, there are many Japanese-made Chinese words, some of which were borrowed by other Sinosphere countries.
In Japan, many words, including 文化, were borrowed from Chinese to give them new meanings as Western concepts, and then re-borrowed by Chinese scholars.
The process is not so one-sided as some would like to present. During Chinese attempts at Westernization, many scholars in China also contributed to and inspired Japanese usage of certain words, sometimes referred to as "Chinese-made new Chinese words" (华制新汉语). For example, among Western books first translated to Chinese during the 19th century were Euclid's Elements, Elements of International Law, and Well's Principles and Applications of Chemistry. Words such as 法律 (law), 函数 (mathematical function), 物理 (physics), 化学 (chemistry), 細胞 (biological cell) were then introduced to Japan. Japanese scholars then made significant contributions in these areas as well, and words were borrowed back to Chinese.