@sfumatosauce I am not sure I understand all your questions correctly.
What I can say is that "network" in this context simply refers to "artificial neural network" which is a rediscovered, now booming mathematical tool used for deep learning. Deep learning is a subcategory of Machine Learning, and Machine Learning is a subcategory of Artificial Intelligence.
Especially for various tasks in supervised learning (classifying images of cats vs dogs, or cars vs people), compression of information (via Autoencoders), and self learning systems ("reinforcement learning"), neural nets often are state of the art.
Now most classic engines like Stockfish, Komodo, Houdini use Graph-Based techniques like Alpha-Beta-Pruning, and evaluate the positions at the end of the search depth according to a long list of criteria, not only simply material, but material adjusted to playing phase (rooks are way more valuable in the endgame compared to the opening, a-/h-pawns are less worth than central pawns in the opening but more worth in several pawn and knight endgames,...), various positional elements like development, bishop pair, doubled pawns, open / closed pawn structures, knight outposts, king safety and so on. So human made criteria are developed by strong chess players and computer scientists, and directly implemented in the chess engine for the evaluation. For these heuristics many parameters need to be tuned, and this is a big part of the development process for future versions.
On the other hand very recent engines like Alpha0 and Leela follow a completely different approach, namely reinforcement learning. Over-simplified they somehow hardcode the rules of the game (in this case chess) and define a win (in this case checkmate), and you decide for a certain structure of an artificial neural net (referred to as the "architecture"), i.e. the number of layers and so on, and you let the engine play millions of games against itself. Now every few hundred games or so the net is allowed to vary certain parameters ("exploration"), and you get some feedback if the change was good or not, and you try to optimize the parameters on the way to win more games against the "older self". After many many games certain weights within the neural net have converged to fine-tuned values. So while the details have some mathematical black box character this net structure somehow manages to learn weights by itself, in a way that if any (so far unseen) position is fed in it can produce a very good idea of how good each possible move is in terms of winning probability (for classic engines this roughly translates to material advantage in number of pawns like +1.5).
So the trained net can be used to assess any position and suggest the best move, similar to a classic engine. However, contrary to the classic case no human heuristics have been used. Also special for the neural net approach is that it can take very long to train a good network, but evaluation is extremely fast. Both use some kind of matrix multiplications, and simply work faster with gpus, up to 100x or some incredible factor.
So as a user, you just download a recent version of Leela, which essentially is a trained neural net for chess. Depending on platform you might need to compile it and make a UCI-compatible chess engine out of it. Then you can use it as a chess engine, i.e. evaluating the current position and the ones within a little search depth.
However, perhaps you were referring to it, you can also help the Leela project with gpu / cpu ressources to build newer networks. You basically use a current net on your hardware to play against itself a number of games, and report back the games. There is a small part of the community that does this, but it's not for the normal user. So I think "network" rarely refers to the people doing this, it'd be very confusing for obvious reasons.
TCEC stands for Top Chess Engine Championship. Basically some people who like computer chess have come up with some rules like time controls, number of games, and so on to host tourneys where engines can battle each other. They also determine the hardware and so on. Some people try smaller projects with their own servers, but TCEC is by far the most established and large-scale project, one could compare it to the olympic games of chess engine "sport".
If you have different rigs but use the exact same version of Leela, essentially they should come up with the same evaluation and move suggestion if you let them calculate until the same depth. Of course some rigs will be faster than others. It would be different if each rig learned the net by itself from scratch since the algorithms have many random components for exploration and initial values. But as I said, you will just use an existing net and use it for evaluation.
And imo gpu rigs just for chess with more than let's say 1x the high end RTX 2080 Ti are only interesting if money has no meaning whatsoever to you ;)
