Running bitcoin-cli commands without hosting a node I want to run bitcoin-cli commands (or JSON API requests) to see example response output in the format as given on https://bitcoin.org/en/developer-reference#bitcoin-core-apis without running bitcoind locally. I do not want to have to download the entire blockchain.

For example, I want to see the output of commands like gettransaction or getblock for transactions and blocks on the main Bitcoin blockchain. Aren't there any publicly available servers that would return responses in the format as specified in Bitcoin Core API? The other blockchain explorers like blockchain.info or blockexplorer.com appear to be returning a response in their own JSON format (i.e. document structure) with different field names and object hierarchy.

I spent quite some time looking for some such service but could not find any helpful information online.
This isn't exactly what you asked, but it's close. 
Take a look at https://chainquery.com/bitcoin-api you can try different JSON-RPC commands through your browser.