(Or why I think the Lorentz transformation is misunderstood)
The speed of light is always measured to be the same in all inertial frames. This is the assumption used to derive the Lorentz transformation visualised in the interactive example above.
While the Lorentz transformation is normally assumed to be a transformation of an event in one inertial frame to the same event in another inertial frame, this derivation indicates that it is not quite the same event, but two events from the same stationary source.
Both events are observed at the same point in space-time along the path of the moving observer. One event is observed by the moving observer and propagates at the speed of light (1 for simplicity) relative to that moving observer. The other event is observed by a stationary observer and propagates at the speed of light relative to that stationary observer.
To see how easy it is to derive this without any other assumptions, take a look at the geometry of the Lorentz transformation.