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.