Facebook documentation on adding the FB Chat plugin to your website seems outdated, at best vague, confusing and incomplete, with missing instructions and information. Follow the simple steps which seems to work for Facebook Pages (not FB usual user profile account, which owns the FB Pages). To date, steps are working and updated for 2024 (March) as of this writing.

Steps to install FB Messenger Chat manually using hard coded scripts, without using CMS plugins or add-ons (e.g. WordPress, Shopify, Wix, etc.):

  1. Switch your Profile to your Facebook Page (not the FB user account which is the owner of the FB Page). You may have 1 or more Profiles depending on how many profiles you have set up for the original main account you are logged in. Alternatively, you may go here: https://www.facebook.com/pages/?category=your_pages&ref=bookmarks to check on the pages that you may have for the account
  2. On the left side menu under Manage Page, click on Settings
  3. On the same left side menu, click on  New Pages Experience
  4. Also from the same left side menu, go to Advanced Messaging
  5. From the Whitelisted domains, enter the URL of your website. Don’t forget to click Add to save.You may add as many as needed. You need to take note of your website resolves to a URL format that has www. or non-www. In any case, always include the HTTPS. The URL needs to be copied and pasted exactly from the URL field when you are visiting or browsing for instance the homepage of your website.
  6. Add the FB Messenger widget to website
    At this point, the next steps seem missing. You have already configured your FB Page profile to permit the domain or your website to use the Messenger widget. Next step is to go to your Facebook Meta Business dashboard. Usually accessible from any of the Meta Business Suite links from any menus from your FB profile. This is also accessible if you are logged in to your profile from the URL as: https://business.facebook.com/
  7. In Meta Business Suite dashboard, from the left menu click on  Inbox. This will show all chats inbox from your users.
  8. On the same chat inbox page, far from the upper right hand corner you will see a gear (settings) icon, click on that
  9.  On the next page, click on Chat Plugin. This will now display settings and configurations where you can actually set up now your FB chat messenger plugin that will display in your website. Do not forget to click on Publish to save. This will now automatically display the chat on your website.
  10. Setup script manually using sample from:
    https://developers.facebook.com/docs/messenger-platform/discovery/facebook-chat-plugin/Example template from FB. You need to replace the PAGE-ID with the actual FB page ID of your FB page, and the API-VERSION. As of this writing, Page ID can be found under About > Page Transparency > Page ID. For the API-VERSION, to date using, replacing it with ‘v6.0’ still works.

    <!-- Messenger Chat Plugin Code -->
    <div id="fb-root"></div>

    <div id="fb-customer-chat" class="fb-customerchat"></div>

    <script>
      var chatbox = document.getElementById('fb-customer-chat');
      chatbox.setAttribute("page_id", "PAGE-ID");
      chatbox.setAttribute("attribution", "biz_inbox");
    </script>


    <script>
      window.fbAsyncInit = function() {
        FB.init({
          xfbml            : true,
          version          : 'API-VERSION'
        });
      };

      (function(d, s, id) {
        var js, fjs = d.getElementsByTagName(s)[0];
        if (d.getElementById(id)) return;
        js = d.createElement(s); js.id = id;
        js.src = 'https://connect.facebook.net/en_US/sdk/xfbml.customerchat.js';
        fjs.parentNode.insertBefore(js, fjs);
      }(document, 'script', 'facebook-jssdk'));
    </script>

    As of this writing, Facebook announced that the Chat plugin may soon go away. Not sure whether the entire widget or just some features as Guest mode, you might want to monitor what happens after May 2024:

    “Chat Plugin is going away soon.
    On May 9, 2024, you will no longer be able to access any functionalities of Chat Plugin. Effective immediately, Chat Plugin in guest mode is no longer available. Other features like m.me links will still be available for you to use.”
    Add Chat Plugin to Your Website | Meta Business Help Center (facebook.com)