CI: Use new documentation previewer
- [X] closes #50832

We've got a documentation previewer service for few months now, but the whole infrastructure powering it was just a PoC not very suitable for the long term (it didn't have error control, was running in a screen session, the workflow wasn't optimal, a github token was hardcoded, it required to remove old previews manually...).

Now we've got a service that IMHO is ready for production. Everything is built independently from pandas in this repo:
https://github.com/pandas-dev/github-doc-previewer/. In the pandas side the only thing we need is to call a GitHub action (check the diff in this PR).

For most people there won't be a significant difference between the previous PoC and the new system, as the UX is the same. Besides stability the main differences are:
- A preview will be generated only for the PRs where it's requested (with the `/preview` comment). Previously we published the previews of every PR.
- The new service is pandas-agnostic. If there is interest, any other repo can use it, the only thing needed is that the GitHub organization of the project is added to a setting in the server. The whole `pandas-dev` organization is of course added. Not sure if we have any other repo where this can be useful, but only adding the GitHub action is enough to have previews enabled.
- The previous point has a small implication. The urls of the previews now contain the org and repo. So, instead of pandas.pydata.org/preview/1234/ the new url is now pandas.pydata.org/preview/pandas-dev/pandas/1234/

Any feedback or problem please let me know.
I'll be merging this soon if there are no objections, so we can use the previewer again (with the new service). Thanks all your hard work here @datapythonista  I had to increase the timeout on nginx, since the requests stays life for a bit more than one minute until the preview is ready and the comment can be posted, but now everything seems to be working fine with the new version.