Handle varied search field configurations

[drammock]

This makes the search shortcut functionality a bit nicer for sites that choose to use a persistent / always-visible search field.

it pulls out some of the work from #777, which suffered from scope creep.

[drammock]

the CI failures here are the prerelease problems with mistune, already reported in #805. Just realized that I forgot to add a special page to the demo docs for testing the persistent search field; will do that now.

[drammock]

OK, I've added a test page to the demo docs... only to realize that it provides an opportuntity for interactive dev testing that was already available on the search results page. So IMO it would be fine to revert 2a925b0 and e11ddb6 but I can also see a case for keeping it (i.e., having a test page specifically titled that way might remind/encourage us to actually check it periodically)

[choldgraf]

This is looking really good to me! I think it is pretty much ready to go

One quick thought - this might be an edge case, but if the user is on mobile and they are on a page with a persistent search bar in the sidebar, and they click the search button, then nothing will happen because the search field will be activated but the sidebar will be hidden already. Maybe that is uncommon-enough that it is not worth special casing though.

[drammock]

That's a very good point. Hopefully it is a rare case and most sites will use the button. But if they don't, this actually feels pretty broken. If like to at least add a warning to the docs about this

[drammock]

OK, here's the rendered docs section with the warning: https://pydata-sphinx-theme--807.org.readthedocs.build/en/807/user_guide/configuring.html#search-bar-search-button

Feel free to push or suggest/commit wording changes if you like

[choldgraf]

This looks good to me! We've already had a bit of iteration on this in #777 so let's merge this in and give people a chance to try it out!