Alternatively, since you've wrapped all of them with this function, you could return a new function from here and not modify the original.

For the scope of this PR, I don't care too much how this is implemented, I just need a correct __qualname__ and changing is good enough.

Yes, you could create a new function, but that's an additional indirection. While negligible from a performance perspective, it shows up e.g. in stacktraces. I'm not even sure that this way of adding functionality to classes reasonable style and what other edge-case consequences that has. Alternatively, one could make the original functions private. But that's all for a different PR.

Do you know examples of each type (Notes/no Notes/no Refs/etc) that we could check here?

For the four branch aternative to detect the insert position:
NOTES: margins
REFERENCES: cohere
SEE ALSO: hlines
NONE: title

Per https://numpydoc.readthedocs.io/en/latest/format.html "See Also" should come before "Notes" and we also have to check for "Examples"?

but it looks like its rendering in the correct order in hlines so I am confused...

It looks like something in the html pipeline is fixing the order as

In [2]: print(plt.hlines.__doc__)
Plot horizontal lines at each *y* from *xmin* to *xmax*.

Parameters
----------
y : float or array-like
    y-indexes where to plot the lines.

xmin, xmax : float or array-like
    Respective beginning and end of each line. If scalars are
    provided, all lines will have the same length.

colors : :mpltype:`color` or list of color , default: :rc:`lines.color`

linestyles : {'solid', 'dashed', 'dashdot', 'dotted'}, default: 'solid'

label : str, default: ''

Returns
-------
`~matplotlib.collections.LineCollection`

Other Parameters
----------------
data : indexable object, optional
    If given, the following parameters also accept a string ``s``, which is
    interpreted as ``data[s]`` (unless this raises an exception):

    *y*, *xmin*, *xmax*, *colors*
**kwargs :  `~matplotlib.collections.LineCollection` properties.

Notes
-----

.. note::

    This is the :ref:`pyplot wrapper <pyplot_interface>` for `.axes.Axes.hlines`.

See Also
--------
vlines : vertical lines
axhline : horizontal line across the Axes

but