probably easier to put links up then try to really explain it but i will give it a high level try
now I am speaking about US citizens on certain things due to regulations
So typically you use an on ramp and off ramp( a service that take US dollars and gives crypto ie coinbase) service to purchase crypto basically you transfer US dollars to Bitcoin or Ethereum etc etc and visa versa. Now these services dont allow exchanges between cryptos and you are limited to the coins they have on their eco system.
Miners just mine and forward their rewards to their wallets, when they want to swap thier coins for others ie mine monero (easier algo less electric consumption) and swap it for ethereum then they go to an exchange that has the trade ie Poloniex Cryptopia Kraken etc etc.
Now the current term swapping is basiclly as you said one coin swapped for another and the two major players in this right now is pancake and uniswap. They are what is considered a decentralized exchange where the previous mentioned are centralized exchanges. Pancake is basically on the binance network and uniswap is basically on the ethereum network. Now once you send your coins to this service; in the app you select your coin and what coin you want and it does all the work for you. Now the claim is that they find the cheapest transfer fees possible then ask if you are ok with the transfer and make it happen.
A few things to note, Binance coin and pancake are nothing more than copied version of the Ethereum and uniswap. Since thier network is not as congested as Ethereum's network, they have much lower gas fees and thus we saw a big migration over to the Binance network and they are making millions
However it is currently not acceptable for US citizens to be on the Binance network. I am not saying that US citizens cant be on it but there is another layer of transaction that is required. Also Binance is so open with its contracts that any contract is acceptable at the moment. The big question here is what happens when other countries regulation make some of those contracts illegal? I dont think anyone can answer that right now.
I hope i helped you a bit, i mean i just talked off the cuff and didn't have a prepared explanation. Be happy to answer as I sure other will be happy to answer any questions you have.