api-docs: Move markdown files to top level directory.

- Updates `.prettierignore` for the new directory.
- Updates any reference to the API documentation directory for
  markdown files to be `api_docs/` instead of `zerver/api/`.
- Removes a reference link from `docs/documentation/api.md` that
  hasn't referenced anything in the text since commit 0542c60.
- Update rendering of API documentation for new directory.

api_docs/index.md

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+# The Zulip API
+
+Zulip's APIs allow you to integrate other services with Zulip.  This
+guide should help you find the API you need:
+
+* First, check if the tool you'd like to integrate with Zulip
+  [already has a native integration](/integrations).
+* Next, check if [Zapier](https://zapier.com/apps) or
+  [IFTTT](https://ifttt.com/search) has an integration.
+  [Zulip's Zapier integration](/integrations/doc/zapier) and
+  [Zulip's IFTTT integration](/integrations/doc/ifttt) often allow
+  integrating a new service with Zulip without writing any code.
+* If you'd like to send content into Zulip, you can
+  [write a native incoming webhook integration](/api/incoming-webhooks-overview)
+  or use [Zulip's API for sending messages](/api/send-message).
+* If you're building an interactive bot that reacts to activity inside
+  Zulip, you'll want to look at Zulip's
+  [Python framework for interactive bots](/api/running-bots) or
+  [Zulip's real-time events API](/api/get-events).
+
+And if you still need to build your own integration with Zulip, check out
+the full [REST API](/api/rest), generally starting with
+[installing the API client bindings](/api/installation-instructions).
+
+In case you already know how you want to build your integration and you're
+just looking for an API key, we've got you covered [here](/api/api-keys).