Indeed, but in my experience many of these places have an unspoken sub-rule that the tattoo rule is not enforced for foreigners.
I know far too many non-Japanese-asian people who get held to the standards applied to Japanese people - not even over tattoos, but things like language, cultural understandings, etc. The aspect of this with white people is where the infamous "gaijin smash" came from.
With Japan and Korea (especially the latter) towards Americans, there's also a soft-unspoken rule (that sort of goes both ways) due to the relationship those countries have fostered towards each other. A Brit/German/Italian/etc who spends more than a short visit in Korea/speaks Korean will probably start being taken to the side for flouting cultural norms like age-deference, polite speech, etc to be informed of their cultural mores (usually phrased with an indication that they also come from a structured society, they should understand that this is the way it is); while this will rarely happen to the same group of Americans. In some cases it's the "dumb/naive American" effect, but it also has to do with the larger relationship between the two countries.
Less importantly but somewhat obvious: many Koreans are fairly competent in English and familiar with common American accents. They’ll know pretty quickly if you aren’t speaking with one.