In theory, we could either have one of us run a mining node for long enough to confirm the transaction or we could find a third-party miner to do it for us on whatever terms we could negotiate directly with them.

In practice, running a mining node without specialized equipment is prohibitively expensive so we're stuck finding a third party. Most of those third parties are looking to do a volume business and aren't interested in one-off small dollar transactions, so it's centralized exchanges or nothing.

No, that isn't how it works at all. You would broadcast a transaction to your friend for $20. It would sit in the mempool and eventually get picked up by miner and would then be forever logged in the blockchain.

So is it possible to broadcast a transaction saying "they paid me X", or is it limited to broadcasting transactions saying "I paid them X"?

Just the latter really. Your unspent transaction outputs (UXTO) would move to your friends address. Bitcoins are really UXTOs.