The second type of kit doesn't need any tools because the button shell has teeth along its inner edge that you use to hook the fabric to secure it in place.
This button style differs slightly from the other in that the shank is a wire that is attached to the button shell and slips through a slot on the button back. To me, this seems a bit more secure because any pulling on the button transfers to the wire shank and not on the back directly. This said, I have not had trouble with the other type coming apart, but for buttons that will be highly stresses, these might be better to use.
The initial step is the same as the first one: you cut a circle of fabric using the template.
Then, you begin hooking the fabric over the teeth on the inside of the button shell. This is easier said than done.
The instructions on the back of the package suggest using a pencil eraser to push the fabric onto the teeth. This does help because the eraser allows the teeth to press into it, trapping the fabric in between.
Be aware that doing this will slightly damage the eraser, leaving some "bite" marks on it, so don't use your favorite pencil for this.
In order to keep the tension on the fabric even, I connected two opposite sides first (as shown here) and then the other two opposite sides.
Continue hooking the fabric to the button all the way around, pulling firmly on the fabric as you wrap it.
Once all of the fabric is secured under the teeth, check the button from the front to check for wrinkles and make sure the fabric isn't skewed or pulled unevenly.
Push all of the fabric inside the shell, then slip the back over the shank and press down until it snaps in place.
As you can see, they both look the same from the front.
After using both of them, I prefer the first method because it is much easier and faster. I will admit that the chances of making a mistake are slightly higher because I have had the fabric shift slightly or wrinkle while putting the back on, but it isn't hard to pop the pieces back apart using pliers or a screwdriver and redo it. If you take a few extra seconds to really make sure that the edges are flattened, you shouldn't have any problems.
On the other hand, the second type does have its advantages. You can see and better control what is happening with the fabric, so if you are trying to exactly place a design, this type would be better to use.
Also, the button form will not accommodate fabric that is not flat. For example, I have seen covered buttons made with fabric that was embellished with silk ribbon embroidered flowers. Because the flowers protrude from the surface of the fabric, it would be better to use the tooth style cover button kit.
I do not like using this cover button type on fabrics that fray easily because pulling the fabric over the teeth puts a lot of stress on the fabric and can shred it in the process.
Helpful Tips for Making Covered Buttons
The metal can show through the fabric as a shine, especially on thin fabrics, so it is a good idea to use a double layer of fabric or line it with a matching neutral fabric to prevent the metal from showing through.
Use a thin piece of batting to give the button some dimension. This gives an especially nice look to buttons used on pillows and other home decor items. What I like to do is to cut the batting smaller than the covering fabric so that it only covers the front and sides of the button. This keeps the batting out of the join between the back and front and prevents any problems with snapping the back on.
Dritz sells a set of plastic cover button templates for cutting circles for the covered buttons ranging from 1/2" to 1 7/8" which is handy to have in case you lose your cardboard template or the box the button forms came in.