baremetrics: Update documentation for current style/structure of integration docs.

[hanyucrocks]

The doc.md for the Baremetrics bot has a few issues compared to the
current documentation style used by other bots (e.g., xkcd, GIPHY):

1. ## Developer Notes section should be removed. This section
   instructs contributors on how to extend the bot's command list. It's
   internal implementation guidance, not user-facing documentation, and
   shouldn't appear in the integrations catalog.

2. Missing "Run this bot" link after Setup. Other bot docs (xkcd,
   GIPHY) include a standard line after the Setup steps:

   Run this bot as described in here.

3. Usage section formatting doesn't match current style. The current
   Usage section uses a separate "Example:" line before the image. The
   current standard (see xkcd, GIPHY) is to put the example inline with
   the command description, like:

   @Baremetrics list-commands - This command gives a list of all
   available commands along with short descriptions.
   Example usage: @Baremetrics list-commands:
   

4. Duplicate word typo. The Usage section currently reads
   short short descriptions.

This was noticed while reviewing
zulip/zulip#38585, which
adds Baremetrics to the integrations catalog and depends on this doc
being in good shape.

[hanyucrocks]

@laurynmm, tagging you for your visibility here. I am working on this issue as said on the CZO Thread.

[hanyucrocks]

@zulipbot add "bug" "help wanted"

[hanyucrocks]

@zulipbot claim

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