DOC: Change core paths to api.typing in URLs
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@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 
Good catch. Thanks @rhshadrach  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 (https://github.com/pandas-dev/pandas/issues/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: https://github.com/pandas-dev/pandas/issues/55632