I don't maintain the old CourseBot (which used to provide an example of a custom integration) anymore.
If you are interested in creating a website chatbot using a custom integration, you can read this article.
However, it has also become progressively easier (since I first developed the CourseBot) to use a plug-and-play integration for Dialogflow and you don't need to develop a custom integration unless you have some very unique need.
Recently, Dialogflow introduced the Dialogflow Messenger integration, which makes it quite easy to add a Dialogflow chatbot to your website. I have written a tutorial on this topic here.
If you are interested in integrating Dialogflow into a live chat service, I would recommend going through this tutorial.