Does anyone have more information about this mystery computer?
Sorry to disappoint you, it's not a computer and not even a computer-like fixed-function machine. It's simply a system of cables, switches, relays and Morse writers. There was no automated signaling. It was just a bunch of switches mounted at pillars along a track, cabled in parallel. Each switch could be pulled by said cable. When a train stopped, the engineer grabbed the cable and pulled. The first pull activated the bell and powered the Morse writer. Further pulls could then be used to send over the position manually. A little clockwork was triggered with every signal; when it ran out, the Morse writer was stopped again. It was built with a mercury switch timer.
Simple but effective by any time's standards - and a great achievement back then.
given it seems electric switches weren't really invented until about 10 years later.
Reliable relays have been around since the 1850s. In theory, they could have been used to make a computer even back then - just at hard to imagine cost :))
I suppose it could have been an "analog" computer of some sort, maybe more like an electric calculator, but even the earliest of those weren't developed till much later as far as I can tell.
There were many mechanical and electro-mechanical special-purpose machinery around at that time.
In fact, 1913 and the Grand Central Station are keywords to this, with the Electric Interlocking Machine installed there. The whole Grand Central system between Grand Central station and Mott Haven (up in the Bronx) was all electric for control and handling of all switches and trains with a block based handling. Much like today. For back then, it was a masterpiece of unimaginable complexity - the whole system with several hundred sections was controlled by just 5 men in a single control room. Including a wall sized display of all sections indicating each switch and its position.
It wasn't the first interlocking system, not even the first electric, but by far the biggest of its time. While for example the Berlin subway system was several times the size, it consists of dozens of independent operating small systems, many of them still mechanical.
While the interlocking system was able to tell an assumed position down to about 1-3 miles, it was independent from above mentioned, way simpler system, and could not narrow the location down. Also it reported to the central switching room, not down to the works.
(Information mainly from memory and an article in Electric Railways Journal of 1914)