Describe the bug + To Reproduce

# has_function.pyx
def f(x):
    return x+1

Cythonize it with cythonize -if has_function.pyx, then:

>>> import has_function
>>> type(has_function.f)
<class '_cython_3_0_0a9.cython_function_or_method'>
>>> dir(_cython_3_0_0a9)
['__doc__', '__loader__', '__name__', '__package__', '__spec__', '_cython_3_0_0a9.cython_function_or_method']
>>> _cython_3_0_0a9._cython_3_0_0a9.cython_function_or_method(  
... )
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
AttributeError: module '_cython_3_0_0a9' has no attribute '_cython_3_0_0a9'

this appears to be general, but it's specifically causing issues with pickling functions in the limited API or with type-slots. I'm not completely sure why it makes a difference to those.

Expected behavior

The function name in the module should be cython_function_or_method (i.e. it shouldn't have the dotted qualname prefix)

Environment (please complete the following information):

• OS: Linux
• Python version 3.8.10
• Cython version: current master branch (3.0.9a ish)

Additional context

It's OK on 0.29.x (you need to change the example to set binding to True)

>>> dir(_cython_0_29_23)
['__doc__', '__loader__', '__name__', '__package__', '__spec__', 'cython_function_or_method']