[@overload]: What is the effect of the *implementation* type-signature?

One thing that I've found underspecified when using @overload is what the type signature of the actual function should be.

For example, let's say I have a function like:

from typing import overload, Literal

@overload
def f(x: int, z: Literal[True]) -> str: ...
@overload
def f(x: int, z: Literal[False] = ...) -> int: ...
def f(x: int, z: bool = False) -> str | int:
    if z:
        return "hello"
    return 1

What is the effect of the last line? In particular:

• The documentation and examples often just use no annotations for the implementation signature (def f(x, z):). Is that the right thing to do (I sometimes get Overloaded implementation is not consistent with signature of overload 1 errors if I leave out the type annotations)?
• If I do specify types on the last, do I also have to add it to @overload list -- in other words:

@overload
def f(x: int, z: Literal[True]) -> str: ...
@overload
def f(x: int, z: Literal[False] = ...) -> int: ...
@overload
def f(x: int, z: bool = ...) -> str | int   # is this overload necessary?
def f(x: int, z: bool = False) -> str | int:
    if z:
        return "hello"
    return 1

Both pyright and mypy seem to interpret things differently with and without it (e.g. see this play link).

I understand that @overload is a complicated feature, but I'm hoping this is a small corner we can start with.

Inizia con gli esempi di @overload presenti in questa issue ed eseguili nel Pyright playground collegato, quindi confronta i risultati con mypy. Esamina come i due strumenti interpretano le firme di implementazione e gli elenchi di overload. Il lavoro è completato quando si è raggiunta una spiegazione condivisa della semantica e sono stati documentati il comportamento previsto e gli esempi.