I am not sure how the link port on the Ti-83 is physically implemented, but:
It might be that you need a crossover cable to get the communication working.
That is if the link port is using a RX/TX scheme, like RS232 (That means that the two calcs sends their data on separate wires).
That means that you might need to swap two of the connections. One of the connections is ground, this must be left as it is.
I'll see if i can find more specific information for you later today.