The key is uniformly accelerating. That wasn't my example at all. A car traveling at constant velocity that undergoes a sudden acceleration is not traveling at uniform acceleration. The coffee just continues at the same velocity in the horizontal direction because the friction forces between it and the dashboard are not great enough to create a differential in the volocity of the cup, i.e. to accelerate it. That is why the cup slides off the dashboard and you have to invent a mysterious fictitious gravity force that emerges out of nowhere because you forgot to notice that the original acceleration was zero.
Non of this changes Einstein's observation of a uniformly accelerating reference frame. You have to meet the condition before the principle applies. If an object in space is undergoing nonconstant rates of change of acceleration with repect to time and you don't know what force is causing the chanes in acceleration and you have no inertial reference frame, then you will be inventing all sorts of fictitious forces to explain these "jerks" in motion.
If the acceleration is not constant, the curvature of the time space is not quadratic but of a higher order, but the same principles apply. Any curvature of timespace can be observed as gravity or acceleration, depending of the frame of reference. It is just that we don't have a supply of massive bodies that loose or gain mass in an observable fashion, hence the textbook comparison is done to a constant acceleration, rather than varying acceleration.
The general theory of relativity is called general because it allows for any frame of reference.