Aren't green threads and async-await orthogonal concepts?

As I understand it async-await is syntax sugar to write a state machine for cooperative multitasking. Green "threads" are threads implemented in user code that might or might not use OS threads. E.g.:

- You can use Rust tokio::task (green threads) with a manually coded Future with no async-await sugar, which might or might not be parallelized depending on the Tokio runtime it's running on.

- ...or with a Future returned by an async block, which allows async-await syntax.

- You can have a Future created by an async function call and poll it manually from an OS thread.

- Node has async-await syntax to express concurrency but it has no parallelism at all since it is single-threaded. I think no green threads either (neither parallel or not) since Promises are stackless?

Is this a new usage of the term I don't know about? What does it mean? Or did I misinterpret the "but"?

As a non-Haskeller I guess it doesn't need explicit async-await syntax because there might be some way to express the same concept with monads?