As a beginner, the safest and kindest way is to do a crest release, which it sounds like your instructor has told you to do. This is where you reach forward up the neck, reins slack, hands resting on the crest. You want your hands down in the neck during the release to give stability and so they don't float. You don't want to lean on the hands, though. There shouldn't be any weight in the hands as your weight should be in your stirrups and leg. It sounds like what you described in the last part is leaning down on your hands which is tipping you forward.
You shouldn't need to be adjusting your rein length on the approach to the fence. Make sure you are organized before heading to the fence, You can always approach with slightly shorter reins, but long arms. Then if there's a need to, you can bring your hands in closer to maintain the contact. Ideally, you should also not have to worry about a horse running out from a fence as a beginner. You should be able to find your straight line and start releasing before the fence, giving you time to find your body and hand position, and the horse should continue on the line. If it's a repeated issue with the horse running out of the fence, you need to ask your instructor to take a few steps back. Either the horse you are on isn't suitable and you need a different mount, or your approach to the fence is incorrect to the degree you are causing a otherwise appropriate horse to run out (ie, turning with too much inside rein, horse was never straight to the fence to begin with) and you need to address those issues before adding jumps to the mix. There's a time and place for a school horse who will run out if not ridden correctly, but that time is not when you are a green beginner.
I suggest you spend some time on the flat practising your release. In your two-point, at any gait, practice reaching forward into your crest release, hold that for a stride, then return your hands back to neutral. This will also help you learn to release with the hands and not the body. You can also do this over poles, using the poles as your 'jump'. It sounds to me like you are getting too overwhelmed with all the other aspects of jumping like steering and speed. Breaking it down should help your body learn the feeling of the release.
I'm not a huge fan of neck ropes for crest releases as I find they are too close to the wither for a proper release. Try grabbing the mane again, even if you can just hook a finger on each hand. If possible, you can do a small braid in the mane where you want the release to give a stronger visual aid and something more solid to hold onto.
Without seeing what the other girl you watched, I can't say exactly what she was doing as your description may not match exactly what was happening. It sounds like one of two things: she was not releasing at all and was incorrect. Without a release, the horse can't use their neck over the fence. When jumping little sticks, it's not a huge effect, but as fences get bigger it will show up in poor form and performance. Plus, it's not very pleasant for them regardless of the height. Alternatively, she could have been using an automatic release where the hands follow the mouth forward so the reins do not slack but the rider maintains contact the entire time. Depending on how much the horse moves for the jump, there might be minimal movement of the hands. The automatic release is more advanced and requires good balance and feel. While it's something to aspire to, you should leave this technique until you are more skilled.