White 1 is the move I learned to play, here in Korea, and that after this invasion white usually has a comfortable result. So black avoids this fuseki in modern go.

This is what I learned in "Becoming 5k" from Baduk TV. However, the teacher in the program seems to believe it still leads to an even game if I understand it correctly.

However, nobody in that lecture except me seemed to agree with that move. They liked the top-right corner approach much better. Of course the corner approach is good too, but than F5 will be a too good move for black. Am I wrong with something?

I don't think you're wrong, but I don't think you're right either. Gasana has already pointed out that this position has shown up in pro matches plenty of times, and the overwhelming favorite play (75% in fact) for white is the approach move. Hardly a ringing endorsement of the immediate invasion, although the invasion is the only move for white that doesn't lose more often than not.

That opening was very popular in the 1990s, there are less than a dozen in the 2000s. This suggests to me that lovelove is right: this opening has all but disappeared for black because the invasion is good for white. Judgements of positions change over time so the fact a shape appeared a lot in the past doesn't mean it is considered even today. If you are happy to play moves than 9ps thought were the best 15 years ago then approach, after all us amateurs are going to make far bigger mistakes later in the game, but it you want to play like a 9p of today then invade.