How did the Federation of Planets communicate and share digital data during ST:TOS? Did the system have a name or designation?
The system of communication across star systems and ships is called Subspace.
Subspace communication (also called subspace radio or the hyperchannel) was the primary form of electromagnetic communication used throughout the Federation. By transmission through subspace rather than normal space, subspace communication permitted the sending of data and messages across interstellar distances faster than the speed of light. - https://memory-alpha.fandom.com/wiki/Subspace_communication
When you wanted to look at the data or system of something remote, you created a subspace link.
A subspace link was a type of subspace communication that connected one point directly to another. - https://memory-alpha.fandom.com/wiki/Subspace_link
For instance, if you wanted to view the records of a colony somewhere, you created a subspace link and then could query their data. This would similar to browsing to a website on the modern internet.
There was a notion of a ship-wide computer network, which seemed to include extensive databases on all sorts of subjects. Nothing in TOS on-screen canon presented, supported, or hinted at the notion of an interplanetary internet.
In the 1960s, most people's idea of a computer was a giant machine tended by scientists in lab coats. It was a stretch to imagine one that could be put aboard a space vehicle and stuffed full of human knowledge. It was an even bigger stretch to imagine one that could understand and reproduce human speech. All this despite the fact that computer technology was developing so fast that much of it was "just around the corner".
One premise of Star Trek is that the ship's mission takes it very far from Earth; so far that even subspace communications could spend days or weeks in transit (it was not instantaneous). Although there were numerous on-screen depictions of lag-less communications (conversations between ship's crew and remote Starfleet authorities), Roddenberry favored the notion of a ship and crew "out there on its own"; this was served by the idea that messages could spend days or weeks in transit. An "internet" as we understand it would not be viable with such latencies - the only information and processing power you'd have access to is what you are carrying with you.