Thank you for all your answers. J.C.MacSwell makes exactly the point I'm trying to make: according to the theory, 1g of ice in the ocean would make the entire ocean remain at 0°C, which it doesn't of course. And I do believe that in an ocean of pure water this would be exactly the same.
I have been thinking it over some more and thought of two extremes, as a thought experiment.
1) Extreme case 1: The heat is applied in one single point at the bottom. In that case it seems inevitable to me that, according to where exactly one measures, different temperatures will be measured. Especially in the case of a very large container with only a tiny fraction of ice remaining (floating at the top of course, to make things worse), when one would measure very close to the heating point, the temperature would be significantly higher than close to the ice. Importantly, no amount of stirring would remedy this (think of the extreme of 1g of ice in 20 000 liters of water, or 1g of ice in the ocean).
So, in case 1, you simply have different temperatures in different areas, even if you stir, so the diagram doesn't really apply in a literal sense. However, it does make sense, see below point 3.
2) Extreme case 2: The heat is applied perfectly uniformly (seems impossible in the real world, but it's a thought experiment). (Maybe the microwave experiment of studiot comes close.) So we have a huge chunk of ice at say -50°C and start applying heat in a perfectly uniform manner (meaning: we add kinetic energy to every molecule at exactly the same rate per molecule). In that case the whole block of ice will turn to water in one and the same instant (in other words: at every point, the molecules will reach the necessary kinetic energy to get loose from the solid state), and then, being completely water, will immediately continue to rise in temperature. (Since the phase change happens in one single point in time).
In this case, the diagram doesn't apply either. You go in a straight, rising line from -50°C to 0°C, then in one single instant (a point in time) you have the phase change, and you continue with a straight rising line. No horizontal part between B and C (in the diagram I posted at the beginning).
3) So to come back to case 1, I think what the diagram and the theory mean, is that the system, as a whole, remains at 0°C during the phase change in terms of the sum of kinetic energies. Temperature is actually kinetic energy, right? (at least in one of the multiple frameworks for understanding temperature). So during the phase change, some areas will have a rather high kinetic energy (close to the heating point, where they will be excited more), say corresponding to 10°C; while other areas will have a low kinetic energy, corresponding to 0°C or even less in the ice; still others somewhere in between, etc. Probably, during the phase change, the total sum of these kinetic energies is equivalent to the kinetic energy corresponding to 0°C.
Does that make sense?
(I propose we leave soup and bishops out of the discussion, since the crux of the question is not really in the "purity" part - see the example of a bit of ice in a gigantic quantity of water).