Einstein's lightening-hit-train thought experiment

"But onlookers from the bank see the forward-facing king sign first because while the light was propagating he and the train moved forward toward the lamp, thus shortening the distance the light had to travel to reach the forward-facing king."

Only if onlookers from the bank see the light traveling at constant speed (independent of the speed of the light source). If, as shown by the Michelson-Morley experiment in 1887, the speed of light does depend on the speed of the light source, onlookers from the bank will see the kings signing simultaneously:

https://en.wikipedia.org/wiki/Emission_theory "Emission theory, also called emitter theory or ballistic theory of light, was a competing theory for the special theory of relativity, explaining the results of the Michelson–Morley experiment of 1887. [...] The name most often associated with emission theory is Isaac Newton. In his corpuscular theory Newton visualized light "corpuscles" being thrown off from hot bodies at a nominal speed of c with respect to the emitting object, and obeying the usual laws of Newtonian mechanics, and we then expect light to be moving towards us with a speed that is offset by the speed of the distant emitter (c ± v)."

http://philsci-archive.pitt.edu/1743/2/Norton.pdf "The Michelson-Morley experiment is fully compatible with an emission theory of light that CONTRADICTS THE LIGHT POSTULATE."

http://books.google.com/books?id=JokgnS1JtmMC Banesh Hoffmann, Relativity and Its Roots, p.92: "There are various remarks to be made about this second principle. For instance, if it is so obvious, how could it turn out to be part of a revolution - especially when the first principle is also a natural one? Moreover, if light consists of particles, as Einstein had suggested in his paper submitted just thirteen weeks before this one, the second principle seems absurd: A stone thrown from a speeding train can do far more damage than one thrown from a train at rest; the speed of the particle is not independent of the motion of the object emitting it. And if we take light to consist of particles and assume that these particles obey Newton's laws, they will conform to Newtonian relativity and thus automatically account for the null result of the Michelson-Morley experiment without recourse to contracting lengths, local time, or Lorentz transformations. Yet, as we have seen, Einstein resisted the temptation to account for the null result in terms of particles of light and simple, familiar Newtonian ideas, and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether. If it was so obvious, though, why did he need to state it as a principle? Because, having taken from the idea of light waves in the ether the one aspect that he needed, he declared early in his paper, to quote his own words, that "the introduction of a 'luminiferous ether' will prove to be superfluous."

The two thought experiments are constructed differently. In one (Einstein's) the "stationary" observer on the platform sees the two events as simultaneous because that's the way Einstein set it up. The thought experiment could have been constructed so that the passenger on the train sees the flashes of light at the same time in which case the platform observer would have observed them happening a different times. Greene's variation is set up so that the observers on the train see the light and sign at the same time. What do they care is some observer on a platform is zipping by outside?

Your final sentence is the key one...

'In other words, if in the first case the forward movement of the train shortened the distance between the source of the light and the onboard observer, why, in the second case, didn't the same thing happen so that the onboard observers saw the signings as sequential, not simultaneous?'

You have assumed that the forward movement of the train shorted the distance that light had to travel. That is only true from the perspective of the onlookers on the bank. For the observer on the train, the light has travelled half the length of the train.

The observer on the train sees the two flashes at different times because they were not simultaneous in his frame of reference, not because the light from the two events has travelled different distances in his frame.

The forward movement of the train does not change distances on the train from the perspective of the observer on the train.