Arrangement of points

Firstly, since there are at least 16 different segments, we must have $\dbinom{n}{2} \geq 16$ , so $n \geq 7$ . Furthermore, we can construct a set of $7$ points which satisfies this:

This shape has coordinates $A(0,0,0), B(0,0,1), C(0,1,0), D(0,1,1), E\left (\frac{1}{\sqrt{2}},\frac{1}{2},\frac{1}{2} \right ), F \left (\frac{1}{\sqrt{2}},\frac{1}{2},\frac{3}{2} \right ), G \left (\frac{7}{3\sqrt{2}}, \frac{1}{2}, \frac{1}{6} \right )$ . Here, we have $AB=AC=BD=CD=AE=BE=CE=DE=DF=BF=EF=EG=1, FA=FC=GA=GC=\sqrt{3}$ , and all other lengths can be verified to not be either of these two, so we're done.