It really depends on how you want to sort them. If you like, you can consider the six simple machines to be types of the two fundamental categories. The fact remains that each of the six requires its own equipment and affects the target mass differently. The lever tends to push, for example, while the pulley pulls (or, to go the other direction, doesn't pull). The screw, wheel, and pulley all go around in a circle, but the mechanism in each case is quite different.
It comes down to having cases where one of those six is clearly the best way to deal with the problem. This differentiates between the actual machines. You wouldn't use a lever or a wheel to get a bucket of bolts six stories up a building under construction. A pulley would be best . . . or an inclined plane if you want your workers to have powerful leg muscles.
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“There must have been a point in early human history when it was actually advantageous to, when confronted with a difficult task, drop it altogether and go do something more fun, because I do that way too often for it to be anything but instinct.” -- Isto Combs
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