"I am creating a simple web application with a free plan. It's an AI chatbot that I would like to make public on my website. I would like to know which plan I should purchase and whether internet users who use the chatbot will be counted as end users, and therefore, if I will have to pay for it."
it depends on the web app. if it's not set as public and you do need authentication then yes users are authenticated using retool (they basically log into retool, then they end up on ur landing page/site and these are the ones that you're charged for as 'end users')
all plans except the free one support public apps with no retool login (no authentication).
with the Business and Enterprise plan (using Retool cloud) you can make Public Apps and/or use Retool Embed. both options allow you to embed your webapp into an external website with or without authentication
you can find the difference between the plans in regards to embeding, using the retool login screen and authentication here
this one shows how u can get the IFRAME code to embed your app somewhere else
The Retool API is available as an opt-in beta for organizations on the Enterprise
Embed a web app with Retool Embed | Retool Docs
bbbuuuttttt.... I'm on the Business plan and I have access to it so 
my best suggestion would be to try using the React-Retool js lib, and see if it works the way you want but I think you'll end up needing at least the Business plan to enable the public app stuff.