To elaborate a little on ApproachMedium's explanation of approach-lit signals...
Approach lighting was originally devised to save energy and to extend the life of the signal lamps. When there was no train approaching the signal, there was no need to keep it illuminated. Many signal installations operated off of batteries that were constantly trickle-charged, so only lighting the signal when necessary resulted in less battery drain and a faster battery recharge.
On heavily traveled main lines, like the Northeast Corridor, signals were kept illuminated constantly because traffic was so frequent that the constant repetitive on-off-on-off operation of approach lighting would have actually resulted in greater wear on the relay contacts that switched the lamp power and would have not saved that much in energy or lamp life. I saw a PRR signal schematic that showed a scheme where signals were constantly lighted unless a loss of AC power caused a switch to battery operation. On battery only, the signals were approach lighted.
With approach lighting, the next signal is illuminated whenever the block before is occupied by a train. As soon as the signal "before" is passed, the next signal will light. If the block is very long or train speed is slow, the signal may light up well in advance of the train arriving. If the signal is near where the train originates it's run, the signal may illuminate as the train comes out of a yard or siding; or when it crosses from track to track in preparation for the return trip.
"What" the signal displays is not dependent on the train in the block "before" of it, but rather the condition and occupancy of the the track "beyond" it. The signal will display the appropriate aspect for the track beyond until the train just enters the block that the signal governs. This is usually less than an engine or coach length past the signal itself. As soon as the block the signal governs is entered, the signal will most likely always display all red: "stop" or "stop and proceed". With 1 block approach lighting, the signal will remain illuminated after the new block is occupied because the remainder of the train is still occupying the block "before". When the rear of the train finally passes the signal, the block "before" is no longer occupied and thus the approach lighting circuit darkens the signal.
Thus it is that the signal will illuminate in advance of the train and display whatever aspect is appropriate for the condition of the track ahead. When the first engine or cab car passes the signal, it will display all red; and as soon as the last car passes the signal, it will go dark.
What I have described is single block approach lighting. I have seen 2 block approach lighting, usually on track signalled in both directions. In this case, the signal heads remain illuminated whenever any part of the train occupies the block on either side of the signal.
As for a "flashing green", a flashing green on a single head or upper head of a multiple head signal does usually represent "cab speed". However, "cab speed" is an aspect that is usually associated with a more sophisticated ( I believe a "9 aspect") signal system that is added to or "overlayed" to the usual 4 aspect signal system. This is mainly done on higher speed lines; typically on the Northeast Corridor. Maybe some NJT Conductors or Engineers can chime in here... is "Cab Speed" typical on any NJT lines off of the NEC?
However, a flashing green on the lower head of a 2 head signal is usually associated with a "limited speed" aspect, either an "approach-limited" or a "limited-clear". These aspects are most likely associated with diverging routes over track switches that will allow the higher "limited" speed rather than the slower "medium" speed. The aspects are similar (red over green or yellow over green), but a steady green represents medium speed but a flashing green allows the higher limited speed.
I believe it was stated that there is a double crossover just past the signal in question. If these are limited speed turnouts and they were routed to change tracks, then a flashing-green "limited speed" aspect is most likely what was observed.
