It's like this:
- 中国人: "a person with Chinese nationality"
- "a person from the Han dynasty"
- "a person of Han (Chinese) ethnicity"
- 華人: "a person of ethnically Han ancestry living outside of China"
- "a person from the Tang dynasty"
- "a Seric; a Cathayan"
The word 中国人 (#1) is by far the most common word corresponds to English Chinese, that is what most lay people naively imagine about the word (which technically tends to be #2-2).
About #2-1 and #4-1, I don't explain here because they are obviously historical terms.
For #2-2, it's good to know that both PRC and Taiwan are multiethnic countries, just >90% of whose population is made up by people who identify themselves as 漢族 "Han ethnicity", or Chinese in a narrow sense. But as you may suspect, this is merely an academic term in Japan that only experts would know.
On top of this, #3 is, in short, overseas Chinese (#2-2) people, or Chinese-hyphen-somewhere people (a.k.a. 華僑【かきょう】).
And for #4-2, did you know those English words? They are older names for Chinese, in other words, you aren't going to use it until you write a historical novel in Japanese.
So, basically what you have to memorize is only 中国人, and for those who have Chinese ancestry but not from China, you should use 華人/華僑 (individual) or 中国系 (group or property). For instance, a Chinese-American is described as 中国系アメリカ人 or 中国系米国人 in Japanese.