There is a difference between הבהמה הנאכלת and הבהמה אשר היא לאכלה (beast of which you may eat): The language הבהמה הנאכלת addresses whether it is permitted or prohibited, while the language אשר היא לאכלה means an animal that was prepared for eating, not only that it is permitted.

And there is a difference between עוף טהור (clean fowl) and עוף אשר יאכל (fowl which may be eaten), that אשר יאכל adds that it cannot have torn flesh, etc.

In order for meat to be kosher, it is not enough that the animal that was slaughtered was a kosher animal. The animal had to be slaughtered and prepared for eating according to the laws of kashrut. The distinction of the language that is discussed in this rule addresses this issue.