Before moving to a Western country, I had a much looser interpretation of vegetarian. In my country I could pick out the meat from a dish and only eat the rest, and that's accepted as vegetarian. If you didn't eat the chicken from a chicken soup, you were eating "vegetarian". This might be because it's a small town (it could be different in big cities), but when I visited a few years ago and my friend returned from Denmark as well for a visit, she ate stews and foods cooked with meat (picking out the meat), but said she is vegetarian. People just seem to have more flexibility regarding diets in some places.

I don't know that much Korean, but they could also have different interpretations of what these words mean.