DOC: Change core paths to api.typing in URLs

[rhshadrach]

• closes #xxxx (Replace xxxx with the GitHub issue number)
• Tests added and passed if fixing a bug or adding a new feature
• All code checks passed.
• Added type annotations to new arguments/methods/functions.
• Added an entry in the latest doc/source/whatsnew/vX.X.X.rst file if fixing a bug or adding a new feature.

@phofl pointed out that #27522 would change the URLs in the API docs from pandas.core to pandas._core. We shouldn't be leaking the locations of the internals via the URLs. While pandas.api.typing is perhaps a little odd, it is at least public.

cc @jorisvandenbossche @jbrockmendel @mroeschke

[phofl]

Thx for addressing this

We might want to add something specifically for docs at some point

[mroeschke]

Good catch. Thanks @rhshadrach

[jbrockmendel]

lgtm

[jorisvandenbossche]

I missed this notification at the time, but remembering it now because it came up in the discussion about deprecating core in the call yesterday.

Two comments:

• This will actually break all existing URL out in the wild. Generally, when we do such a renaming, we add redirects for the existing pages (we have that mechanism set up, so it's just a matter of adding all impacted pages redirects.csv, AFAIK)
• Given that changing urls is a bit annoying, we might want to give the final location a second thought. If we think about "we might want to add something specifically for docs at some point" to quote Patrick above (DOC: Provide a public place for users to link to our documentation #55632), we might better to that now, before those urls start to be used in the stable docs, to avoid breaking the urls a second time

I am all for breaking those urls once so we can remove the "core" out of those long references (which we need to deprecating core anyway, but those long references to "pandas.core.groupby.<..>" in the public docs have also always bothered me).
But personally I am not sure if pandas.api.typing is the ideal place to put those.

I will move the second item about what's the best location to use for the docs to the issue Patrick opened: #55632