@Rover From what I understand, it also depends on how strong the fire is. You are correct that wet wood has a really hard time catching on fire. However, once it's on fire the fire simply needs to burn stronger and bigger than whatever is trying to put it out. I suppose what's important to note is that lightning has tendency to pierce whatever it hits, and in this situation the lighting pierced a tree, therefore setting the inside of the tree that is still dry on fire. Once the inside caught on fire, it grew big enough and strong enough to the point where the wet wood and the rain didn't matter as the heat would eventually dry the outer wood before burning it. This is why forest fires can still happen during a thunderstorm(like in this chapter) despite how strong the rain is, and also why it's so hard to put the fires out once they become too big to handle as the fire can simply dry whatever wood becomes wet before spreading. Often times the only way to stop a huge forest fire is to isolate the fire by chopping down the trees in it's path(fire usually follows the wind) that haven't caught on fire yet, thus preventing the fire from spreading any further. Otherwise, you just have to let the fire burn itself out on it's own, usually due to the wind direction causing the fire to burn towards some obstacle that can't be burned or away from the rest of the trees and towards the edge of the forest, or when the wind completely dies down and the fire can't reach any neighboring trees without the wind's help. With all this in mind, it's nice to know the author did some research on forest fires as the elves had actually mentioned chopping some trees down so as to prevent the rest of the forest from burning, even at the expense of more homes being destroyed. Thankfully, however, Douglas was there to save the day.
Again, I'm not an expert on all this and I'm pretty sure there's a lot more factors that come into play, like how close the trees are to each other, the size of the trees, how strong and/or dry the wind is, and the overall quality of the wood, so take what I've said with a grain of salt.