Recall what happens when Anakin kills Dooku, which I paraphrase as follows:
Palpatine: Kill him.
(Anakin kills Dooku)
Anakin: I shouldn't have done that, it's not the Jedi way.
And that's your answer: it's not the Jedi way. This whole scene is showing you how much Anakin has fallen into being Vader, even claiming he (Vader) killed Anakin (himself). By contrast, Obi-Wan shows how much he has not fallen from being a Jedi, and how he has risen from his weakened and defeated state he started the show in. He has a defeated, wheezing, essentially defenseless Vader in front of him. But he will not kill someone in that state, for that is the Jedi way.
The scene also suggests that Obi-Wan still sees a glimmer of Anakin within him. The blue/red lighting changes in the scene suggest a switching persona between the old, good Anakin (blue) and the new, wretched Vader (red). Obi-Wan acknowledges that he cannot save Anakin himself, that when it comes to him there is nothing but Vader, but he can perhaps also see that Anakin is still there to be saved. It would just have to be done by another.
Allowing himself to be pulled in to killing Vader just sullies himself and draws him closer to the dark side, which serves no good purpose. Vader may go on to be a scourge, but he is the puppet of another. Obi-Wan trusts in the Jedi way and the will of the Force. An act of "benevolent" murder may save some immediately foreseeable troubles, but what new ones unlock as a result? Who steps up to fill the void? Could it be Obi-Wan himself? Will Obi-Wan be in no (mental) condition to guide Luke towards the redemption of Vader and the defeat of the Empire? Perhaps the Empire could not be defeated at all, for it was not Luke who took out the Emperor but Anakin. Let go of this attachment to vengeance and what you think you know is coming soon, and you can see and be guided by the Force to something greater and further out.
As far as the line "This ends today. Either he dies, or I do", recall that in the Clone Wars series Satine describes Obi-Wan as "a bundle of half-truths and hyperbole". Granted, Satine was in an agitated and argumentative state at the time, so not inclined to be flattering, but the description is nevertheless quite consistent with Obi-Wan's character. Obi-Wan frequently does make use of half-truths and hyperboles, e.g. "from a certain point of view", in order to motivate others to the ends he wishes to drive them. Or, at times, just to amuse himself, as he also has a certain degree of curiosity—in said series he also delays rescuing another Jedi because he wants to see which orifice a mind control worm is going to enter, and bets on it with Anakin—and a penchant for snark.
As such the line in question may just be his flare for the dramatic kicking in to help him convince the other person to accept that he's going to leave and bait Vader away. He could always argue later, as he does in the OT, that "Vader killed Anakin" is true and satisfies his claim that one of them would die. Or just other metaphorical deaths, such as the sort of psychological death Obi-Wan starts the series in; or the tortured state of having lost again and having no choice but to go on indefinitely suffering from it for Vader.
Finally, note that Obi-Wan completes his reconnection to the Force and the Jedi ways during his battle with Vader. Reconnection to the Jedi ways reinforces the point this answer starts on, and reconnection to the Force reinforces the subsequent point: he may get an instinctive impression from the Force that the long run is better served by letting Vader survive, so that he may later be redeemed and the Empire overthrown. So he may have started with "one of us must die", only to feel in that moment that in fact both of them must survive.