Soap and rain water, why does it not rinse off? I am in Papua New Guinea at the moment, where rain water harvesting is the most common way of getting water.
When washing hands, the soap feels hard to rinse off, and my skin feels "greasy".

I read around a bit here and here and I think I understand what is said: essentially, sodium stearate in the soap separates into sodium and stearate ions, which then precipitate with other ions (like Calcium, into calcium stearate) and get rinsed away. But in very soft water, there are sodium ions rather than calcium ones, so this reaction is slowed down a lot, resulting in the greasy feeling.
As it turns out, I have access to hard ground water at the same time, and the difference is very obvious: with hard water, soap is rinsed off instantly.

My question now: I assume rain water has in fact almost no ions in it (I'm thinking Sodium ions mostly), and I would therefore assume that the Sodium stearate dissolves very easily, but has no replacement ion to precipitate with, therefore sticks to the skin.

Is this the right explanation or are there other factors at play in the explanation about that greasy feeling?
Yes, this is a good explanation. Rain water is almost pure water, it lacks of bivalent ions such as $\ce{Ca^{2+}}$ and $\ce{Mg^{2+}}$ which helps soap to rinse of. You will get the same effect when a water softener is installed on water distribution.