Aside from the high-level similarity of the "function -> state machine" transformation, Rust's is quite a bit different (and IMO both simpler and more flexible).
A C++ coroutine chooses a particular promise type as part of its definition. Its frame defaults to a separate heap-allocation per coroutine, with some allowance for elision. At a suspension point, it passes a type-erased handle to the current coroutine to an `await_suspend` method, which can either `return false` or call `handle.resume()` to resume the coroutine. A stack of `co_await`ing coroutines (or "psuedo-thread" as you call it) is thus a linked list of `coroutine_handle`s stored in the coroutine frames of their await-ees, rooted with whoever is responsible for next resuming the coroutine.
A Rust async function does things inside out, in a sense. It has no promise type; calling one directly returns its frame into the caller's frame, as a value with an anonymous type that implements the `Future` trait. This trait has a single method called `poll`, which resumes the function and runs it until its next suspension point. `poll` takes a single argument, a handle which is used to signal when it is ready to continue. This handle is threaded down through a stack of `poll`s (a "task" or pseudo-thread), and stored with whoever is responsible for notifying the task it should continue.
One implication of the Rust approach is that the "executor" and the "reactor" are decoupled. An executor maintains a collection of running tasks and schedules them. A reactor holds onto those handles and notifies executors of relevant events. This lets you control scheduling without language hooks like await_transform- you can associate your prioritization data with a task when you spawn it on a particular executor, and it can await any reactor without losing that information.
Another implication is that you have a choice of whether to a) `await` a future, making it part of the current task, or b) spawn it as its own task, to be scheduled on its own, much like OS thread APIs. Option (a) can get really interesting with multiple concurrent sub-futures (with things like Promise.all or select); it can be as simple as having the caller poll all its children every time it wakes up, or as complex as wrapping `poll`'s handle argument and implementing your own scheduling within a task.