Charles Hoskinson indicated in this tweet that he favors the term "programmable validator" to the term "smart contract" with regard to Cardano.
Quoting the tweet Charles Hoskinson is responding to, "In Cardano, "smart-contracts" tend to refer to the combination of the on-chain validator(s) and the off-chain code that's driving it / them."
Here's what I've gathered so far. (This is speculation. Please do not believe it and please do correct it in the answers and comments.)
All operations (and data) necessary for Cardano to function are executed (and stored) on 3rd party-owned compute resources known as "validation nodes" (with the caveat that Cardano itself owns a number of "validation nodes"). The results returned by these 3rd parties can be trusted, statically speaking, thanks to redundancy -- each operation is performed by all nodes in a random sample of validation nodes, and results are checked against each other.
This also describes code running "on-chain" and AFAIK is similar to how contracts execute on Ethereum.
All blockchains have "off-chain" code driving the miners or validators.
Are all these statements correct? If not please correct me and if so, what's the big difference between Cardano contracts and contracts on other chains alluded to in the above tweets?