There seems to be two ways to gap from what I have seen.
The first is how I would say most people do it, get to the edge, preload, get height, extend legs and land. There isn't anything wrong with this technique and it's handy if you need to go over something, but on larger gaps it can make the landings pretty harsh. This way the rider usually twists the bike which can make some landings awkward and less accurate.
The second is how some of the comp guys to it (Stan Shaw being one of the best I have seen) where you start off exactly the same, but instead of getting height, you let the bike stay level or in extreme cases (Like Stan) even let the back wheel go lower than the level of the gap. I've seen this technique get some impressive distances. It's a little odd as you actually gap perfectly straight, it also has the bonus of SUPER smooth landings and really good control and accuracy. Not so great if the gap is low or you need to go over something though.
Laurence, you are doing the first technique which is fine, it does look like you need to pinch off the edge even more than your second vid shows. That gaps not small though, probably bigger than I can do now so all I can say is practice and it will come.