According to this Wikipedia page, for the Beltrami-Klein and the Poincare disk models, their unit disk has the same coordinate as an Euclidean unit circle with range and domain -1 to 1.
However, when I apply the Cayley-Klein metric, with C = 1 for the Poincare model, I get a different distance than one in the Cinderella program.$${\displaystyle d(a,b)=C\log {\frac {\left|bp\right|\left|qa\right|}{\left|ap\right|\left|qb\right|}}}$$
I was able to fin $|qp|$ to equal 0.619 rather than the 1.47 as in dicated below.
Idea points B and A are (-0.99,0.16) and (0.32,-0.95) respectively. And points q and p are (-0.76,-0.03) and (0.06,-0.73) respectively.
What concept am I missing or flaws in my understanding?
In another question of mine I'm discussing how the coefficient relates to the curvature in different models. That would explain a difference of a factor of $2$ between your result and that of the Cinderella hyperbolic measurement.
But your difference is actually not so round a number. So something else is probably amiss.
From where did you get the coordinates of the points? Are they supposed to match the screenshot? Cinderella is using the Beltrami-Klein model behind the scenes. So even if you are looking at a Poincaré model representation, the points on a geodesic would be on an Euclidean line. Which I believe they are, as far as I can tell with the few significant decimals you have included.
But even if you were to do all the computation with Beltrami-Klein coordinates by accident, your result should only be wrong by a factor of $2$ (due to the incorrect choice of $C$). Could the rest be rounding errors? Can you try again with more decimals shown for your points?