Since a lot changed within the last years, I wanted to make an update on what's new in the WordPress webhook world (2019+).
The document you mentioned is provided by WordPress.com, which is a different service than the self-hosted websites via WordPress.org.
To archive that goal there are a couple of possibilities:
1. Use an already existing plugin
The easiest way would be to use a plugin that already offers webhooks. If you want to cover a specific use-case, it makes sense to search for a specific solution. For example, Woocommerce offers slack integration as a plugin: https://woocommerce.com/products/woocommerce-slack/
In case you want a global solution so that you can basically integrate everything, I recommend you WP Webhooks - They have a free version available and it works great out of the box. It allows you not only to send data from WordPress but also to receive data. You can also connect it easily in both ways with Zapier, automate.io and a lot of other services.
Another possible plugin would be Hookpress - It also offers you the possibility to send data from WordPress to any webhook. The only issue is that it is not maintained anymore and there are compatibility issues with PHP 7+
Another plugin would be WebSub/PubSubHubbub - It works great as well but has more specific use-cases.
2. Custom Code
In case nothing of the above suits your use-case, you can always start creating your own piece of code for it or hire a developer to do it for you. In case you want to do it by yourself, you should check out the following WordPress functions: wp_remote_post and wp_remote_get
If you want to hire a developer, I recommend you to ask on fiverr or Upwork or in case you want only expert developers Codeable
If you know something I don't know, feel free to reach out and I'll add it as well :)