How, as a technical matter, do miners express their desire to fork or not to fork? What happens with the miners that are in minority if they persist in mining on the unpopular branch?
From my understanding if you just use the Official software you cannot decide to fork, because adding blocks to the blockchain is done automatically. If you do decide to fork I think you need to modify the source code and put some extra logic on how, where and when you want to add a new block to the blockchain.
For the second part of your question: if a couple of miners decide on forking, what they basically do is create a new blockchain that runs in parallel with the old one.