I am finding this intriguing, you are opening up a whole new world to me.
I have never understood, nor seen any indication of, hyperbolic geometry in relation to Minkowski Diagrams.
Yes, I have seen the hyperbola in Minkowski's diagrams in his Space and Time Lecture but had no idea that that led to a completely different geometry...
So perhaps you can understand why I have been so reluctant to let go of the Euclidean perspective.
But are we not still dealing with the same equations?
Isn't the Minkowski hyperbolic treatment just one way of depicting them, because the hyperbola seems to be associated with losing the uniformity of scale on the diagonal lines, which in turn gives rise to what I see as anomalies: planes of simultaneity and jumps on the time axis of twin paradox diagrams.
Don't get me wrong I am not saying there is anything wrong with what you are saying, only that it is intriguing how far I have gone down the wrong road because I have failed to appreciate that hyperbolic geometry is integral with Minkowski diagrams.
For example this from the introduction to Minkowski Diagrams
Wikipedia said:
Minkowski diagrams are two-dimensional graphs that depict events as happening in a
universe consisting of one space dimension and one time dimension. Unlike a regular distance-time graph, the distance is displayed on the horizontal axis and time on the vertical axis. Additionally, the time and space
units of measurement are chosen in such a way that an object moving at the speed of light is depicted as following a 45° angle to the diagram's axes.
Nowhere can I find any suggestion that it involves a different metric (if that is the correct term?)