The Japanese word for tattoo is 入墨 irezumi and is used to refer to Japanese-style tattoos. Because of the strong relationship with yakuza, they will get you banned from public baths or from my golf club.
I’ve got a number of Western friends with tattoos, and some have had problems in public baths or pools and others have not. Small tattoos won’t and foreigners visiting over seem to be OK. I don’t have any Japanese friends with tattoos, so I don’t have personal experience with that combination.
I’ve seen a few Japanese women with small tattoos. Mostly this was in Western bars and occasionally outside, but usually in places where they could be covered up easily. You can see a few Japanese men with tattoo.
Japanese will use タトゥ katakana pronounced “tatu” for Western style tattoos. Here are some tattoo magazines. I have yet to see a Japanese person using kanji as a tattoo.
I don’t know that much about tattoos, but my assessment is:
“They are copies of popular tattoos seen in other magazines” → Likely.
“The phrases are copied and pasted from pages on the internet, and then assigned a gothic font.” → Or from Japanese to English dictionaries.
“The tattoos were done in the US.” For a small percentage of people, but not many.
“The tattoos were done by American-born artists in Japan.” Again, possible but not too likely.
“I suspect the real reason is that English, both written and spoken, is better understood in Japan than Japanese is understood in the US (especially among tattoo artists).” Actually, it’s probably about the same percent chance, e.g., close to zero that you would get fluent speakers in either category. You’re more likely to find tattoo artists in Japan who trained in the States than vice-versa.
English is much easier to write than kanji, especially for non-natives of the respective languages.
And to answer the question in the OP. No, you do not see many Asians running around with English tattoos, and I’ve yet to see another language used.