Yes, Williams "Jim and the Indians" and the drifter problem (the usual name for what you describe as the five transplant recipients vs. one drifter case) are philosophically distinct and generally considered different.
Working from memory, that's also a difference Williams is aware of when he comes
up with it.
In the drifter problem, the agent is killing someone to save others without the threat that they will all die. = 1 person or 5 other people
In the Jim and the Indians problem, the agent is killing someone under the promise of another agent that doing so will make it so the agent doesn't kill everyone. = promise of only 1 person or threat of that 1 + 19 others.
I think your description
It seems that both experiments are equivalent, they are the case of killing one life by one own's hand vs letting many die by refusing to kill.
misses several relevant distinctions which I think amount to substantial differences:
- whether the person we kill would be included in the list of those who would otherwise die
- the Jim and the Indians example contains a threat. is that they will kill all the Indians if you don't kill one. I don't see an equivalent feature to the threat in the normal version of the drifter example.
- the Jim and the Indians example contains a promise. The leader has promised that everyone else gets to go free if you just kill one Indian. Conversely, the drifter contains a stipulation that the drifter is a perfect match for the five.
Your intuition depends on believing the threat -- in other words, if we accept that they will kill all 20 if we don't kill 1, then we aren't saving even one life by not killing. Conversely, in the drifter case, we are killing 1 life to save other lives.
Conversely, one might suggest not killing in the Jim and the Indians context precisely because one does not believe the promise or believes that one cannot trust the promises of people who are willing to kill.
Yes, a Kantian rejects both, but this is because on both universalization formulas and humanity formulas, killing itself is wrong -- since we cannot will the end of our own and all rational life as a universal and we cannot see killing as an action that furthers humanity and treats it as an end. In other words (with the exceptions of war and capital punishment for reasons that involve the nature of justice and the state), Kantian ethics does not allow agents to kill for any reason whatsoever.
For Utilitarians, the drifter case is a commonly presented problem. And "Jim and the Indians" is to some extent an improvement on the enticement of the problem for Utilitarians. In Mill's version of Utilitarianism, there's a harm principle and a liberty principle that prevent the distasteful result that we should kill drifters if it would help save the lives of upstanding citizens. The Jim and the Indians version (partially) fixes this by making it so that the sacrifice in question will (probably) die anyway.
The feature that causes the parentheticals, however, is the use of the threats and promises. It requires the agent to be willing to accept that the captain will keep his word and actually save the other 19 or conversely kill all 20.
These features actually point to a common issue for utilitarian views -- namely, whether we are trying to maximize expected consequences or actual consequences. If we are trying to maximize expected consequences, then it seems like if believe the Captain, then we ought to kill the one person (expecting to save 19 lives). If we mean actual consequences, then we can only know if killing the one person was good or not based on whether or not the 19 end up getting saved.
Part of Williams' point is that it seems intuitionally off (at least for him and many) to think that killing someone is the right action regardless of the threat/promise. Or to put it another way, it's part of Williams' larger critique that Utilitarianism and its calculus is not the basis of our moral action but rather than an attempt at redescription that we then evaluate based on whether or not it gets things right.