[Using simplifed Chinese:] 中国 means "China". 汉字 means the Chinese characters. (and I think also Japanese and Korean) 中文 and 汉语 both mean "Chinese". 中文 (with the same 中 from 中国) is more used for written Chinese and 汉语 (with the same 汉 from 汉字 which refers to the Han people) is more used for spoken Chinese. Most of the time you can use either. When we say "Chinese" in English, it could mean many things. So for the language it is 中文 or 汉语 (普通话 for "Mandarin Chinese"). For a Chinese person you can say 中国人, for the food 中国菜, etc. Sometimes 中国 or 中国的 can mean Chinese too. There are other words too like 国语 and the Chinese ethnicity and so on which might just be "Chinese" in English.

汉语 means "Chinese" as Chinese language 中国 means "Chinese" as belonging to China

It is so good that you know the differences between 汉字 and 中文. Besides, 汉语 stand for a language ,and 中国 refers to a country.

We call Chinese language 汉语 in general. 汉字 means the chinese spelling characters. As for the word Chinese itself, it has various meannings. Generally Chinese means things belongs to or come from China. For more uses it also refers to its language and literature. “中文”,"国语”, "汉语”, “中国话”, basically these are all deferent express ways of Chinese language or literature.

In language adjectives and nouns relating to most ethnicities or countries have these variants - referring to the country the people the language the culture; the Chinese variants are if anything less ambiguous than the English equivalents - consider 'an English teacher of Chinese studying English as a second language' and see how many different interpretations it might have. Seems in English we don't distinguish the language and the nationality so often. The 国 in Chinese words of nationality is a dead give-away, as is the 语 or 话 in names of languages