It's good you feel your mid back light up, that is key. I'm highlighting this because pull-up strength does not really translate over to snatch and jerk strength except in the most general "back gets more fit" kind of way. It's kind of like saying your squat carries over to a lateral lunge. Maybe...but the movements are so different that the carry over is not significant except in the "legs are strong enough to work" kind of way. Specific firing patterns and recruitment is key to movement specific strength.
Leaving aside technical considerations and just thinking about physical strength, rowing strength is a better indicator, and overhead squat strength better than that. RDL strength and snatch grip DL strength are also indicators.
Your shoulders or triceps might be the weak link too, given that you say that presses have always been a challenge for you. But the shoulders should not be taking the majority of the emphasis on any of those movements. The back is what gives supports strength, which is why I mentioned it in my first post. I'd suggest working on both really, not suggesting you are wrong necessarily. Just another angle to work on.
Squat strength is not the sole determinant of snatch or jerk strength either. Your legs are plenty strong if you're hitting the squats deep. One of the things that makes me believe your back is weak is precisely that your squat is so big in comparison to your olympic lifts. That can either be due to technique problems or weaknesses in muscle groups.