Nihonbeijin or Nichibeijin?

Alright, its probably a weird question.
Either when I was in High School or just starting University (late 90's - early 2000's) trying to learn Japanese and it came to nationality I had a hard time answering. (I'm Nisei 二世 but at the time I didn't know that term.) I couldn't say Nihonjin 日本人 because I'm not from Japan. So, I'm Amerikajin. But Amerikajin didn't quite feel right either. Then someone told me the word I was looking for was Nihonbeijin 日本米人, Japanese-American. Okay, Yes! a word that fits! I was able to look it up on the internet and the definition actually showed up. I was excited. Fast forward many years later, still trying to keep some of my language skills and randomly looked up Nihonbeijin. :( It's gone. Still works in the ol' translator but it's gone. Then I find the word Nichibeijin 日米人which is supposed to mean the same thing.
So, the real question is which one is the right one to use? And did we 'lose' 日本米人 in favor of 日米人 because its shorter one character?

I can't recall ever hearing someone use the term "Nihonbeijin," and Googling for 日本米人 doesn't bring up any results for the term being used in Japanese writing either. Plugging 日本米人 into Google Translate does spit back "Japanese American," but I have a hunch that's simply the program trying to force fit a translation.

On the other hand, the term I usually hear used in Japan to refer to Japanese Americans is "Nikkei Amerikajin" (日系アメリカ人), which also has a page on Japanese Wikipedia.

Generally speaking, -jin is used to denote nationality, while -kei is used to show ethnicity. For example, as a Lebanese American, I'm "Arabiakei Amerikajin" (the one time I accidentally introduced myself as "Lebanon-jin) to the owner of a Lebanese restaurant in Tokyo, he immediately started speaking to me in Arabic). Ordinarily, this would make "Japanese American" "Nihonkei Amerikajin" in Japanese, but since "Nikkei" exists as a set term, it becomes "Nikkei Amerikajin.2

「I'm Nisei 二世 but at the time I didn't know that term.」

While I know Nisei is a common term among Japanese communities in the U.S., I don't think I've ever heard anyone in Japan use the term. On occasions when I use it when speaking in Japanese to people in Japan, I usually have to tack on an explanation, and Googling for the term in Japanese brings up multiple sources that mention using it to mean "ethnically Japanese" is something done in countries outside of Japan (i.e. people in Japan don't use Nisei to mean "Japanese").

The reason why is that technically Nisei just means "second-generation," so using it while talking to someone in Japan will have them wondering "Second-generation what?" And even if you explain that you're using it to mean you're ethnically Japanese, referring to yourself as Nisei would also imply that your parents were born and grew up in Japan. If, on the other hand, your grandparents were the last generation of your family to grow up in Japan, you'd be "Sansei," and the term changes again the farther back you go.

Of course, if you're speaking English and use the word Nisei, the fact that it's not an indigenous English word gives it a special, and Japanese, significance, and since most English speakers aren't versed in Japanese vocabulary for numbers, Nisei has become the catch-all term for "ethnically Japanese but not living in Japan." For people in Japan, though, Nisei doesn't have that connotation/expansion, which is why "Nikkei [country]-jin" is the more common term for Japanese speakers in Japan.

「someone told me the word I was looking for was Nihonbeijin 日本米人」
It's possible whoever told you that was mistaken. Linguistically, "Nihonbeijin" is a strange-sounding construct because it uses the full form for "Japanese" (Nihon) with the short form form for Amerika (bei). While that doesn't break any strict grammar rules, it's generally not done since it makes the phrase feel sort of unbalanced and unnatural.

Ah, sorry, you also asked about Nichibeijin. That's also a term I can't recall anyone in Japan saying, and like with Nihonbeijin 日米人 doesn't show up in any Google results either.

Linguistically, it also doesn't work to convey "Japanese American" because without -kei, there's nothing that specifically indicates ethnicity. While Nihonkei Amerikajin indicates "Japanese ethnicity, American nationality," "Nichibeijin" sounds like it's trying to say that a person's nationality is an even mix of Japan and America.

As such, if you referred to yourself as Nichibeijin to a native Japanese-speaker, I think they might take it as a slightly poetic attempt to say that societally/culturally you consider yourself to be internationalized in half-Japanese, half-American way...or they might just be confused and not understand you.

I agree that the term you are looking for is “nikkei amerika-jin,” American of Japanese descent.

While the two terms mentioned “might” have been used among certain communities back then (particularly in the States), I have never heard of those, and I agree that “nisei” by definition simply means “second generation,” and while it is widely understood among communities, for example in California, where there are many “nisei” and “sansei,” etc., so they would use that term to distinguish among themselves, it might not ring a bell immediately if you use that term to Japanese people living in Japan.

Neither of them are correct, and I'm happy to learn that the Internet became smart enough to deny at least one of them.

The word you're looking for is "Nikkei-Amerika-jin". It means "Japanese-American" and is written as 日系アメリカ人.

You could also call it "Nikkei-beikoku-jin" 日系米国人 which is less common but still works. On the other hand, we never say "beijin" to mean "American".