How to integrate dialogflow with website?
Dialogflow doesn't provide any direct way to integrate the agent with your website. As robert mentioned in his answer The web demo integration really is just for demo purposes and not customizable. If You want to integrate dialogflow agent in your website you have two options:
Dialogflow provides APIs and SDKs for integration. You need to integrate these APIs in your website. You can refer to this article for more information on this.
Another way is to use any third party tool which provides Dialogflow integrations. In my opinion, Kommunicate provides smooth Dialogflow Integration with a set of Actionable Messages. This article might be helpful for you.
The web demo integration really is just for demo purposes and not customizable. To do a real integration with your website with custom UI, you call the “detect intent” API from your server and build your own UI around it.
See https://cloud.google.com/dialogflow-enterprise/docs/reference/rest/v2beta1/projects.agent.sessions/detectIntent