If the limb and flesh are dangling off a live person, they are tahor (ritually pure), and they do not transmit any tum'ah (ritual impurity) of a corpse. If the person then died, the dangling limb is still tahor because it is considered like flesh that is separated from a live person, which is tahor, very similar to the last Mishnah. We did not elaborate much on this in the last Mishnah, but we view the dangling limb as if it came off a second before the person died, or, in the last Mishnah, before the animal died. Therefore, it is like a body part that came off during a person's life, and it is not tamei (ritually impure). It is not like the person; it is not part of the corpse anymore because it came off a millisecond before the death, and it is not part of that body.
The same applies to the last Mishnah: when the animal just dies and you have a dangling limb, we view it as if it came off right before the death. The dangling limb does still transmit some sort of tum'ah, in terms of tum'at met (corpse impurity), but only as long as it remains a limb. But if a bone or some flesh separates from the limb, they are not tamei, because they are no longer part of that limb. This is the opinion of Rabbi Meir, who says that flesh or a bone that separates from a limb of a corpse remains tamei. If it separates from a detached limb of a live person, it becomes tahor. Rabbi Shimon declares the flesh or the bone to be tahor when it is separated from the limb in either case: if the limb was from a corpse or from a live person. It does not matter.