The interaction pops up correctly, but the attribute doesn't bind to script:
I don't have the webchat embedded on any page, the system invokes the chats programmatically upon certain external events, so I can't rely on the browser API explained here: https://developer.mypurecloud.com/api/webchat/
Unfortunately, this isn't possible at this time. Creating a 3rd party chat using POST /api/v2/conversations/chats adds the prefix context to your attributes. This makes your input of scripterFooVar1 get set as context.scripterFooVar1. Scripts require input variable names to match participant attribute names exactly, and cannot contain periods. Due to the way the attribute is set and the limitation on script variables, there is no direct way to use an attribute set by that API as an input variable in a script.
As a workaround, you'll need to use a custom Data Action in your script to make API requests to access the participant data on the conversation and set that data to script variables.
I've added your request to CC-2857 which exists to support this feature, but is not currently scheduled for work.
On the development team we took at look at the lift to get input and output variables to play nice with 3rd party chat variables. After some looking yesterday we believe we have the code in place to handle this case now. I am looking to deploy the code into the test environments before the holidays and then in the early part of January get it deployed to production for use. We just need to get testing around this before hand.