I am assuming for this answer that you are wanting to adjust commercial reeds made from cane (not synthetic materials), and that you are not trying to make your own reeds from blanks or from stalks of cane.
I have always felt that my time with the saxophone was too limited as it was. So, I always wanted to keep the reed work as simple and quick as possible while getting the desired results. It so happens that I also have spent relatively little money doing so.
Also, it is important to store reeds well so that they last. Doing so is, in my opinion, at least as important as adjusting the reeds correctly.
With all that in mind, here is a list of what I use to deal with reeds. It's really not as extensive as it may seem at first glance.
Materials and tools
- Good reeds to begin with. "Good" depends on what you like, of course. As a classical saxophonist, I've never found anything I like better than the standard Vandoren traditional "blue box" reeds, but the techniques apply no matter which brand you are using.
- Proper containers for the reeds themselves once they are used. The plastic cases that the Vandoren reeds come in are not good for storing them (explanation below). I still use the old La Voz (now just Rico, I think) ReedGard-style cases, but there are lots of other products out there that I presume would be just fine. Some people prefer the type that have ridges under the reed to let air in. I have some of both type and honestly haven't noticed a difference in the reeds stored in each.
- Some kind of container for the reed containers. The idea here is to manage humidity. (There are products that combine this function with the reed cases themselves, though I have found the present solution to be more cost-effective.) I've had decent luck with just a plastic Tupperware-style box-shaped container. The lid can seal or be left a little open, depending what I need to do with the humidity.
- Some kind of humidity regulation for the container. A ball of rock salt (maybe an inch or so in diameter) held together by a nylon can work nicely for this. You don't want the container to be too dry (which causes problems with how the reeds play, see below) or too wet (which can allow the reeds to decay and mold). Some of the commercial case/container combinations use charcoal. I've never tried charcoal.
- Wet/dry sandpaper, 320 or 400 grit. This is one example that should work. I used to be able to get something like this from the automotive section at Wal-Mart for about $4 for a package, but I haven't bought any in some time. You can cut the sheets into smaller pieces for convenience.
- A truly flat surface on which to work. Some people like plexiglass. I've always been fine just using a table.
- Optionally, a reed knife. You can get by without it if you have the sandpaper, and I did exactly that for a long time. It is quicker to have one, though, for the balancing adjustment I describe below. For single reeds, I don't think it matters what shape of reed knife you have, though I suppose it should be comfortable for you to use. I think I bought mine for under $20, but that was a few years ago.
At least in the American Midwest, especially in the winter months, the biggest problem with reeds is warping due to uneven drying. I'm not referring to the wrinkling at the tip of the reed, but rather a more "baseline" change in the reed such that the table (flat side) of the reed is no longer flat. A warped reed will not seal against the mouthpiece and, therefore, will be harder to play than a flat reed, and it may be tough even to get a sound from the reed. You can check for warp in two ways. If you are playing, try taking the mouthpiece off the instrument while keeping the reed on. Cover the round hole of the mouthpiece with your palm, then suck the air out of the mouthpiece. If you can remove your mouth from the mouthpiece while the reed stays closed on the mouthpiece, eventually producing a "pop" sound as the reed comes away from the mouthpiece, you have a good seal. If you cannot make that happen, or the pop happens immediately and is very weak, you probably have a warped reed. The other way you can tell is to take the reed off the mouthpiece and put it on a flat surface. Try rocking the reed from side to side. If the reed rocks that way, it is warped. The farther it rocks, the more warped it is.
Now, when you play, you want the reed to be wet. We hope that getting the reed wet cures at least some of the warp issue, and in fact, that is your best bet: if a reed is warped, try soaking it in water for fifteen minutes. I use something to hold the reed underwater in a cup of water. (Specifically, I use the plastic holder that the Vandoren reeds ship in.)
If that does not work, you can try sanding the warp out. Put your sandpaper on the flat surface and rub the reed back and forth across the sandpaper, taking care not to sand the area anywhere near the tip. This may work, but it is really difficult to get the back of the reed truly flat this way.
One other method is similar to something described in the first video in the question. Use the reed knife to try to get the reed table flat again. Again, this is difficult to do well. The factory will usually do a much better job at getting the reed table flat than I can, so I usually try to avoid working on the table unless the reed is otherwise a lost cause.
Balancing and strength adjustment
If the reed seems too hard, or you have problems playing is a certain range, the reed may be out of balance rather than specifically too hard. The second video above shows a test on clarinet for balance, though it is not quite clear from the video itself what he is doing to check. What I typically do on saxophone (alto or lower) is to turn the mouthpiece on the neck so that my lip is holding one rail of the reed shut, then try to play in the low range down to the lowest note. I turn the mouthpiece the other direction and check the same thing. If I can play without difficulty on each rail, the reed is probably good in this range. If the rails play differently, the reed is probably out of balance. If the rails play the same but are both too hard, the reed is too hard.
If the reed is out of balance, you need to remove material from the side that is harder. Remember which side was left open to vibrate during your balance test when you had more trouble playing. Using the knife if you have it or the sandpaper if you don't, remove material from the "low-register" area. The following diagram shows an approximation.
You will want to remove a little material, then test again. Especially with the knife, material comes off faster than you might expect. If you are using a knife, it is important to have the knife moving in a back-and-forth scraping motion prior to contacting the reed; otherwise, you risk gouging the reed.
If the reed was just too hard, you can use sandpaper over the area of the vamp (on the non-flat side of the reed). Use just a few strokes, avoiding the reed tip. Then, check again.
Once you have both rails playing the same in the low register, balance the middle register and high register using the same procedure, removing material from the approximate area you need.
Note that if the reed is too soft, there is little you can do to adjust it. You can try a reed trimmer if you want, but you will not get a lot of extra life out of the reed that way. You can also make the reed seem a little harder by positioning the reed closer to the tip of the mouthpiece.
While there are certainly more reed adjustments available, some requiring more specialized equipment, I have found that procedures above solve nearly all my reed issues.
Evaluation of the videos
You asked about the Youtube videos, so I will say a few words about each.
The knife technique for flattening reeds is one I also use, but rarely, because I find it very hard to do as well as the factory in getting the reed table flat. I do not use the knife on the reed table for any other purpose.
Seriously, what's with the hood? Anyway, I am skeptical of the device being advertised for reasons similar to those I give in #4 below. Maybe it would work, but I do not feel like spending the money to try it, and I certainly wouldn't try it with any kind of blindfold.
The rest of this DVD may or may not have something useful. It looks like it's going the right direction, but without seeing the rest of the DVD, it is hard to comment.
This device has three problems that I see. First, it is possible that your mouthpiece is not exactly symmetrical, so a reed that is exactly symmetrical will not play as a balanced reed on your mouthpiece. Second, cane is not an internally consistent material. Just looking at the reed, you can see that there are thicker and thinner fibers. They are not guaranteed to be in any equal proportion through the reed. So, there is no guarantee that a "perfectly"-shaped reed will actually have uniform resistance throughout the reed. Third, I have no way of knowing what shape the device actually produces, but the first two problems are sufficient that I would steer clear of this device. On the other hand, maybe someone else would have good luck with it.
I would be happy to attempt to answer any questions. It is very difficult to explain some of these ideas in a web forum, and I may have been less than clear in some of my prose.