[over.oper.general] says

The co_­await operator is described completely in [expr.await]. The attributes and restrictions found in the rest of [over.oper] do not apply to it unless explicitly stated in [expr.await].

we only mention operator co_await operator in [expr.await] p3

Evaluation of an await-expression involves the following auxiliary types, expressions, and objects:

• [...]
• o is determined by enumerating the applicable operator co_­await functions for an argument a ([over.match.oper]), and choosing the best one through overload resolution ([over.match]). If overload resolution is ambiguous, the program is ill-formed. If no viable functions are found, o is a. Otherwise, o is a call to the selected function with the argument a. If o would be a prvalue, the temporary materialization conversion ([conv.rval]) is applied.

Except that, we didn't mention operator co_await in [expr.await] any more. In the complete subclause [expr.await], we never give the specification that which scopes an operator co_await can be declared in, what number of parameters the function shall have, and how the lookup rules perform to find the function. We never state these details in [expr.await].

struct Awaiter{
    bool await_ready(){return false;}
    void await_suspend(std::coroutine_handle<>){}
    void await_resume(){}
};
struct CoroutineType{
    struct promise_type;
    using handle = std::coroutine_handle<promise_type>;
    struct promise_type{
        auto initial_suspend(){
            return std::suspend_always{};
        }
        auto final_suspend() noexcept{
            return std::suspend_always{};
        }
        auto get_return_object(){
            return CoroutineType{handle::from_promise(*this)};
        }
        void unhandled_exception(){

        }
    };
    handle coro;
};

// contrast 1
struct C{
    Awaiter operator co_await(){return Awaiter{};}
};
CoroutineType fun(){
    co_await C{};
} 

// contrast 2
struct D{};
Awaiter operator co_await(D){
   return Awaiter{};
}

CoroutineType  fun2(){
   co_await  D{};
}

// contrast 3
namespace X{
    struct ADL{};
    Awaiter operator co_await(ADL){
        return Awaiter{};
    }
}
CoroutineType fun3(){
    co_await X::ADL{};
}

By testing the implementations(e.g., Clang and GCC), it seems that having a different scope in which operator co_await is declared, the function should accompany by a different number of parameters. As well, a different lookup rule will perform to find the declarations.