In terms of async, it's when you have to have a function with "async" attached to it and making it so that only other async functions can call async functions.
It ends up creating a weird circumstance where you can end up with a lot of duplicated APIs, particularly in libraries, because you are providing both async and non-async versions of functions.
For instance, if you resolve a future in the wrong context you'll still have problems - the coloring is just a compile time error that you are doing things wrong, rather than a runtime deadlock.
[0] https://journal.stuffwithstuff.com/2015/02/01/what-color-is-...
The terminology is used to talk about languages that have async and sync functions where you declare (or color) the function as either async or sync.
In these languages it's pretty common for the language to enforce a constraint that async functions can only call other async functions. Javascript / Typescript, Python are popular examples of languages with colored functions.
In this context: functions anotated with async
I think what they're trying to say is that Java's green thread implementation has special support for async I/O. Threads that block on I/O aren't polled for completion by the runtime, instead they use OS async features under the hood.
This allows Java's green threads to compete performance-wise with async/await solutions, but with cleaner code that doesn't need colored functions.
In older green thread implementations in other languages, I/O can actually cause significant CPU overhead due to polling threads that are blocked by I/O requests.
"Stackful coroutines" allow yielding from any arbitrary point. They're basically fibers, except with the explicit control transfer and value passing yield and resume operators; there's no hidden or implicit control transfer like with green threads. Though, some people would argue allowing any function to yield without announcing this in their type signature is tantamount to hidden control transfer. Personally, I don't see how that's different than allowing any function to call other functions, or to loop, but in any event languages are free to layer on additional typing constraints--constraints that can be tailored to the desired typing semantics, rather than dictated by implementation details.
Stackless coroutines are typically implemented as a special kind of function whose state is allocated and instantiated by the caller. In contrast, stackful coroutines are typically implemented by reifying the stack, similar to threads. The "stack" may or not be the same as the system's ABI stack.
In stackful coroutines, unless there are additional typing constraints imposed by the language for hygiene reasons, any function can typically be called as a coroutine or use yield and resume. There's no need to compile functions into special alternative forms as call frame management works the same whether invoked from a coroutine context or not.