Answer

These discussions have come up here more than once. My claim is not about what the medieval authorities or later authorities write. It is well known that they bring various practical differences. But in my opinion, none of these is actually a practical difference for the conceptual inquiry of whether it is permitted or overridden.
The connection to the question whether to slaughter or feed carrion is a misunderstanding. The discussion whether the Sabbath is permitted or overridden in the face of danger to life is not about the Sabbath specifically, but about prohibitions in general. Therefore, if the Sabbath is permitted, then eating carrion is also permitted. And if this one is overridden, that one is overridden as well. So there is no such practical difference at all. As far as I remember, the consensus of all the medieval authorities who dealt with this is that it does not depend on the question of permitted or overridden. Each one gives reasons one way or the other, but I do not recall anyone who makes it depend on permitted or overridden.
And if you take the question whether one should minimize prohibitions—choosing the lesser prohibition first—that is indeed the main practical difference brought by the medieval and later authorities, but truthfully, even that is not really a practical difference. Even if the Sabbath is entirely permitted, what is permitted is only what is needed for the rescue. What is not needed was not permitted. Therefore one must minimize prohibitions as much as possible even if one understands that the Torah is permitted in the face of danger to life, because if it is possible to minimize, then you are not violating the prohibition for the sake of the rescue, but just for no reason, and that certainly was not permitted.
As for the practical difference of ending the Sabbath before the time of Rabbeinu Tam, if you think it is possible to end it, why not do so? After all, as I wrote above, even according to the view that it is permitted, there is no point in violating a prohibition if there is no need. True, one could perhaps say that there is a cost to ending the Sabbath before the time of Rabbeinu Tam, and therefore maybe it is better to remain in the Sabbath and violate it for the sake of a sick person. But you would have to define what the cost is of ending the Sabbath before the time of Rabbeinu Tam, so that this practical difference could be discussed. Simply speaking, if ending the Sabbath earlier is valid, then there is no such cost. The Sabbath has ended, and that is that. In short, it depends on the question whether the Geonim are right or Rabbeinu Tam is right, not on the question of what you should do in practice.