It must be that the figure with the half circles is just a representation of the hyperbolic space into 2D. Such projections are not faithful; you cannot take measurements in the projection and take them literally.

We can make an analogy to cartography: you can't trust areas and distances on distorted projections like Mercator.

Look, even the angles don't look to be zero in that diagram. We have to imagine that we zoom in on an infinitesimal zone around each corner to see the almost zero angle; i.e. the circle tangent lines actually go almost parallel. So to speak.

Thus the angles are locally correct, since they are measurable on arbitrarily small scales and can easily be imagined to be even when glancing at the entire figure. But distances between the points aren't localizable; they have to follow a measure which somehow correctly spans the abstract hyperbolic space that they represent.

How about this (almost certainly incorrect) imagining: pretend that the real line shown, on which the three points lie, is actually a horizon line, which lies in a vast distance (out at infinity). Just like the horizon when you do drawings with two-point perspective. Imagine the three points are vanishing points on the horizon. Vanishing points are not actually points; they just directions into infinity.

if, in a two-point perspective, you draw a curve whose endpoints are tangent to two vanishing point traces, that curve is infinitely long.

For instance if you draw an intersection between two infinite roads, where the curb has a round corner, you will get some kind of smiley curve joining two vanishing points. That curve is understood to be infinitely long.