It's not my area but from what I've heard schools that don't require qualified teacher status, like yours, are usually classified by those in the business as 'fake' international schools. In order to work at a 'real' international school you have to have QTS and usually a couple of years' expereince in your home country. They don't teach EFL at these 'real' internatioal schools, or if they do they employ a teacher already teaching a subject at the school. As I said, international school reddits can give you more info about this. If you're teaching EFL full time you're an EFL teacher, so your job would be considered part of the TEFL industry in Korea.
Wow these guys sound... uhh... "fun."
Yes, unsurprisingly they greatly look down on everyone working in the TEFL industry. A few of them pop up on rtefl fairly regularly, doing similar kind of schtick as Koreaboo.
Just make sure you don't work at an "International School" that isn't actually an international school. These places want to hire native English teachers who aren't going to be fluent in Korean legalese or familar with the reputation of a place in a country they've never been to before. https://www.koreaherald.com/view.php?ud=20170709000269