Chullin 37: When is a Tereifa Not a Tereifa?

Today we learn a new Mishna about what we are permitted to sacrifice.  It teaches that when one slaughters an animal that might die in the near future,

• Rabban Shimon ben Gamliel says that the slaughter is valid only if the animal convulses with all of its legs following the slaughter
• Rabbi Eliezer says that it is valid if blood spurts from the neck
• Rabbi Shimon agrees with Rabbi Eliezer: it is valid if it is slaughtered in the evening and the next day the walls show evidence of blood spurting
• The rabbis says that it is valid only if its legs convulse or if it wags its tail

These indicators apply to animals both small and large (ex. a sheep and a cow) who are in danger of imminent death.  They do not include small animals whose forelegs move briefly which is "part of the natural course of the removal of the animal's soul from its body and not a convulsion which demonstrates life".  If the animal is healthy and not facing imminent death,  the slaughter is valid even if none of these indicators is present.

The Gemara discusses the positions of these different rabbis.  They speak to how one might determine the status of an animal ready for sacrifice.  When is an animal a tereifa and when is it simply ill and permitted for slaughter (within the above limitations)?  When is such animal actually piggul?  When is the animal absolutely forbidden and when might it be permitted?

After so much attention has been paid to the disqualification of a tereifa as sacrifice, it is somewhat surprising to learn that a dying animal might not be considered a tereifa.  Perhaps the rabbis were aware of the limitations of the people, and they hoped to allow people to benefit from the mitzvah of sacrifice. Or perhaps there is another reason for this leniency.