I need to know which one is the correct forehand grip. As you can see, in the first image the area beneath the thumb is not pressed completely(only a little bit against the handle most of the time(not even during the tightening of the grip before contact) and the bottom leftmost palm area is also a bit free unlike in the second grip where both of them are touching the racket. If you need to know more I'm happy to help! (I've verified the grip is correct from the other angle, makes a sharp v and is on the same line as the racket head. So which is the forehand grip? Here's a small video too with better views 'imgur.com/a/gFcI0g7'
Well lets assume everything else is correct and the only question is the gap. For neutral forehand grip, I think the first picture is correct because it allows for easy transition to other grips. For forehand strokes, I think there are some strokes where the first picture would be correct, and the some strokes where the second picture would be correct.
It'd be interesting if @SSSSNT elaborated on which types of forehand, but here's a thought. I don't think the following is necessarily important re the thumb bend, but I wouldn't ever have my thumb bent so sharply. It doesn't seem natural.. it seems tense, when your grip is meant to typically be somewhat relaxed. (and also some personal preference 'cos if the thumb got bent even a tiny bit more then it'd be injured! If you wanted to punch somebody then sure the thumb would be bent that much, to keep it out the way, but I haven't seen a need for such a sharp thumb bend when holding a badminton racket) . In your first pic your thumb is bent 90 degrees. The other one your thumb looks less bent, which i'd prefer as a choice. . Another aspect is the angle between the racket and the forearm. This is something you mention regarding the difference between the two pictures, though you don't call it that. So, in your first picture, where the butt of your racket is closer to your forearm, you have less of an angle. And in the second picture, where the butt of your racket is further from your forearm, then it's more of an angle. And I suppose when the racket shaft is at more of an angle then the thumb has less contact(you mention re thumb contact). I think maybe for a late wide forehand that isn't too far out, then you might have more of an angle. Whereas for a late wide forehand that is far out then you'd have less of an angle. So for example, if you have a forehand that is very wide, then your racket head has to point out to the side , and you will have less of an angle(between racket shaft and forearm), and the shaft will have more contact with the thumb. Whereas if the forehand is wide but not so wide, and you are still able to point your racket head upwards (which is beneficial 'cos it allows lots of forearm rotation into the shot), then you'd have your racket shaft at more of an angle relative to your forearm. Maybe there may also be some wrist deviation too to adjust the angle of the racket shaft, and gravity can play a role too in where the racket naturally sits. I may not be entirely correct but that's some thoughts. Open to correction.
