It is now August 2020, and a big chunk of this article was written in 2018 and 2019.
Should you still create your own custom integration even in August 2020?
I think the answer to that is: probably NOT.
Here is the step by step process I would use to decide how to integrate a Dialogflow chatbot into my website today.
If your requirements are STILL not satisfied by any of the above options, then yes, you should use a custom integration.
If you still decide to read this tutorial, I would be very curious about your website bot requirement in that case. In mid-2020, it is almost entirely unnecessary to build a custom integration for your website.