Gook

As pointed out in another post some people may think that waygook is considered a racial slur. It's not quite that, but after some quick research it turns out that the two words are incredibly related. Both Wikipedia and The Urban Dictionary have a similar explanations; here's a brief description from Wikipedia's definition of gook:

An Asian person, especially an enemy (e.g. Koreans or Vietnamese during the Korean and Vietnam wars). By extension, any Asian person. Probably derived from the Korean words “hanguk” and “miguk”. “Hanguk” refers to Korea and “miguk” is the common word for the United States. American troops thought "miguk" sounded like "me gook" (i.e. "I am a gook"). The word persisted during the Vietnam War, perhaps also because the Vietnamese people have a similar word “quốc”, meaning "country".

So the term gook only derives a derogatory meaning in a North American context. In Korea it's the word for country (국) and since it describes a person's nationality everybody in Korean ends up being a kind of gook.

The Korean word for foreign (외국 aka waygook), or foreigner (외국인 aka waygookin) or even foreign person (외국사람 aka waygook saram), all use 국 as a base word and the use of gook in this context is so far removed from North American meaning that nobody recognizes it as a derogatory term. So when you have a website for foreigners in Korea, waygook.org seems quite appropriate.

As for spelling, the competing romanization systems create many transcriptions for 국 including gook, guk, kook, kuk, etc. Even the language tools disagree, making no discernible difference between waygook or wayguk.