It is easy to scratch or gouge an LCD. You should be careful when cleaning one. Unfortunately, fixing a scratch or gouge is not easy.
As you note, a number of solutions are online. And most do sound scary.
Scratches are most likely to affect the LCD's screen coating. So, repairs often involve polishing down part of this coating. Some repairs apply a new coating to part of the screen.
Many people have reported success with Vaseline. This does not remove any coating. Rather, the Vaseline fills the scratch, making it less noticeable.
Basically, you clean your monitor. Then, apply a dab of Vaseline. Next, remove the excess. The idea is to leave a small amount in the scratch.
I would try this method first. It seems to be the least risky.
However, be careful how you apply the Vaseline. Use a clean, dry, lint-free cloth. Avoid paper towels and toilet paper; they could scratch your monitor, too.
If Vaseline doesn't work, there are other possible solutions. For instance, a pencil eraser might work. It should be soft and clean (no graphite smudges). I'd buy a new one. Gently rub the scratch with the eraser. Some have reported success with this method.
Next up is auto rubbing compound. You apply it to the scratched area of the screen. Then, rub it with a soft, clean cloth. After the scratch is removed (along with the screen's coating), apply clear lacquer. (I didn't find information on the lacquer, or where to get it.) Personally, I wouldn't do this. You could make the problem worse. Or, you could introduce new problems.
A better solution is an LCD scratch repair kit. These kits work much like the rubbing compound. Novus, Merlin's and Displex are brands to consider. A kit runs about $10. Read reviews online before buying.