even wood that's been kept covered is often very wet and can be
difficult to light.
I will second that. Some people claim their wood is dry, but if you go look at how they "cover" it, sometimes you see that the wood is only protected from the lightest of rains in the lightest of winds. I have seen some "coverings" that any amount of normal wind would blow the rain right under and onto the wood.
Techniques for fire starting in wet conditions
What are some techniques for lighting wet firewood?
Get at the drier wood on the inside
Try to dry out the firewood as best you can, if that is possible. Either way, make sure that you cut into the wood; as long as the wood is only wet on the surface, you can get drier wood on the inside.
To get at the drier wood, you can use a bladed tool (generally knife or hatchet) to cut off the wet outer part. Also make sure you split the wood as much as you are willing to do; the increased surface area will help a lot toward getting the wood to catch fire.
Use lots of tinder and kindling; as much as you can make
Even more importantly is to use a lot of tinder and kindling. When I light in wet conditions, sometimes I spend most of my time making a pile of wood shavings with my knife. I make enough of them that I have a reasonably large pile to light, and plenty more to keep adding if the larger pieces are having a hard time catching. Have extra tinder on hand for the same reason, in case the shavings themselves are too moist, or in case the fire goes out and you need to re-start it.
On top of your tiny kindling, keep working your way up to slightly larger pieces; do not suddenly jump from small kindling to very large pieces of wood.
Some methods for more extreme wetness
If you are trying to start a fire while it is currently raining, as opposed to after a rain, it can be even more difficult. I have found that the humidity and rain are sometimes bad enough that no amount of kindling will do: the tinder and kindling are becoming too wet simply by being in the wet air. When your resources are this wet, you need to use something that catches easily and burns hotter, like fat-wood, magnesium, or super-fine steel wool. You can even combine these things: my father likes to use magnesium and fat-wood together.
Fat-wood is a naturally occurring kind of wood which is saturated with resin. You can also make a home-made variant by soaking wood in flammable resins. It catches a spark or flame relatively easily, and it burns hotter and longer than normal wood. You then continue as before, working your way up with larger pieces of wood, once this has helped you get your start.
Take some very small shavings off of it, and some larger shavings. You can even make toothpick-sized kindling and bigger if you're having a hard enough time. Usually, you make tinder and kindling from it then put what's left of your fat-wood piece away to help with more fires later, but if you need the fire fast or bad enough, or if it's a short trip or have plenty to spare, you can put what's left of your fat-wood stick in over your fat-wood tinder and kindling since whole fat-wood sticks help too.
Magnesium burns super-hot; many thousands of degrees. The magnesium for fire starting comes in a small block, and you scrape some of it off the block to make a small pile of scrapings/shavings. Then, usually a spark is applied by using a ferro rod; as long as the spark lands directly on the tiny magnesium shavings it should give you a super hot spot for a very brief time which you use to get your damp tinder and kindling lit.
Steel wool, the super-fine grade stuff, burns quite hot too. I once saw someone completely submerge it under water, take it out and shake it off for a minute, then drop a spark on it with a ferro rod. He then blew on it for a few seconds, and it was hot enough that kindling on it starting catching.
When conditions are very poor, my preferred method is to use charcoal. If I can get even a tiny spot on the charcoal to glow red, I can then blow on it to increase the size and heat of the red spot, then I hold a second piece of charcoal up to it so that it too can start getting hot; once the two of them are working together, it is easy to keep working up from there, possibly with even more small pieces of charcoal if the conditions are bad enough.
This technique requires a lot of blowing. Be prepared to be exhausted after. Also, once it starts to smoke, try to turn away quickly to breathe, and be prepared to accidentally inhale some smoke that will choke you out for a moment.
The up-side to this technique is that the charcoal burns hot, and you can keep forcing it to stay hot by continually blowing on it; so much so, that I have multiple times skipped kindling entirely and jumped straight from small pieces of charcoal to larger pieces of wood. Of course, lots of kindling is still helpful, and it will save some of your breathe. When all else fails, I get the initial charcoal red spot by applying a bic lighter to a sharp corner on the piece of charcoal.
Counter intuitive method: use green wood
This next one sounds completely counter-intuitive, but I have used it successfully. This one is back to wet conditions after a rain, not during the rain, and it only works if you have a nice, sunny day after the rain.
Putting objects out under a bright sun, especially if it's mid-day, can help dry things out a lot better. Smaller, thinner objects are going to dry out even faster, and this is good for your tinder.
Grab up a thin, live (yes, still green) branch from a tree. Scrape the bark away (if it is a kind of bark that is good for starting fires, then obviously use it as well), then start scraping at the inner wood. Try to keep the stuff you scrape off the wood to be as thin as you can possibly make it. I find it works best for me if I hold my knife perpendicular (straight up and down) on the wood and just scrape side to side, instead of whittling into the wood. Sometimes I even use the 90-degree corner on the back of my knife instead of using the actual blade, and I scrape this back and forth along the inner wood of the thin branch. I do this for a long time, building up as large a pile as I can.
These ultra-thin shavings are so thin that, by the time I am done, the first half of what I worked on is much drier. But now it's time to make them even better yet; go spread this tinder bundle out in the direct sunlight for 10 to 30 minutes to dry them out even more. Because they are so super thin, they dry out very fast in the sunlight. That is why I use green wood instead of dead wood for this part; I have found that I can get the shavings even thinner than I can with dead wood. Once it dries out in the sun for a quarter hour or half hour, now it's dry wood and it's super thin which is great for tinder.
With green wood, I can get the stuff I scrape off so thin that sometimes it looks fuzzy instead of woody. And since it is so thin, which is good for lighting, I have a few times gotten it to light after a rain even without the sun-drying, though don't skip the sun step if it's shining.
Yes, this sounds crazy and counter-intuitive. I saw a guy do it on YouTube and thought "He must be crazy for choosing green wood," until he explained it and actually got a fire lit with it. I had to try it to make sure he wasn't pulling our leg, and it worked great!
Whatever you do, make sure you have multiple alternatives available to you. I keep multiple tools in my pocket-sized tinder box. It has a tiny piece of fat-wood, a few scraps of char cloth, a few tiny pieces of charcoal, a small chunk cut off of a magnesium block (spot reserved, it's not currently there), and a small ferro rod removed from its handle. Sometimes I have a tiny tinder-nest in there too from the afore-mentioned scrapings off a thin, green stick.
With all of those options, it is rare that I need anything else to get a fire started even in wet conditions.
Your other question
Are there ways to tell wet pre-cut firewood from wood that's only damp
on the outside?
Try to push into the wood. If it is only surface-wetness, you should not be able to push into it at all; the wood should feel completely firm and tough as usual. The wetter the wood is, the more you will be able to push into it.
For example, I just did a fire recently where I did use some very wet wood. I had some branches that I could push into about a quarter-inch, more in some spots. I scraped this wettest part away and burned the rest; it did not burn well, but the fire was already established, so it did eventually burn.
For wood which is wet, but not wet enough to feel soft at all, you could try looking for signs like fungus. Not all fungus will be large mushrooms; even smaller spots, which could look almost like mold (and might indeed be mold).
The location of the wood is another indicator. If the wood was flat against the ground, it is more likely to be wetter all the way through. If it is not flat, maybe a branch that is curved enough that part of it is held off the ground, or one side is propped against a tree or on a bush or another branch, or if the entire thing is caught up in something and not on the ground at all, etc.: since it has been kept off the ground, it is likely to be drier on the inside.