I have read all your comments and I will think about the the proposed solution. Unfortunately, I was not able to reply to them earlier. As usual, thanks for your comments and time. Regarding this sentence in my post: Is the subset $P^{*}_{2}(A)$ reduced to at most a unique partition? Yes, the expression at most is essential for my question. Obviously, the existence is not guaranteed for dimension $k>2$ and for $|A|=4$. On the contrary, I really did not understand where the adjective pairwise should have been used or not.

True. That's very true. The problem has a convexity nature more than a linear one. I start to think that such a configuration simply does not exist. I think that the proof provided by @JanKyncl. It seems that Jan uses a Caratheodory convex argument to prove this. Then I suppose that there is some chance to have such desired configuration when |A|/2 > 3 and |A| is even. In the case made explicit in my question. If |A|=<3 then the desired configuration seems not to exist, i.e. |A|=6, then 3=3.

@GerryMyerson Hello Gerry, you are right. I mean that the six aren't all on the same line. However, an example where any 3 points of them are not collinear would be better. Such a more general example should show more the generality of this lack of uniqueness. However I am reading your example and I am checking if it is true.

Thanks. Don't know if that picture is so robust against my question. I mean the nice counterexample is based on the fact that there are pieces of the curves with a curvature is equal to 0. In the extreme case, two straight lines (curvature 0) can meet for sure infinite times. For sure I was very imprecise in posting the first question that is appearing now. I don't think that the second question is totally trivial.

@ToddTrimble Yes I agree with you. I thought that the new question would have been posted as a fresh question. I think I have misunderstood the following fact: My question has been considered off topic but it has been answered. For sure it is my fault. My apologies.