dayschedule/dayschedule-widget ? reverse-engineered prompt

Build me an embeddable appointment booking widget for a website, with both a popup version and an inline version. It should let a visitor pick a time and book meetings for one on one, round robin, or group scheduling, and support meeting links for Google Meet, Zoom, and Microsoft Teams.

I want it to be easy to drop into any site with a simple script tag or CDN link, and to have a nice API like opening the popup from a button click or rendering the calendar directly on the page. Make it customizable so the colors can match a brand, including a dark mode option, and let me hide the header or event details if needed. It should also support pre filling booking form questions.

Please make it fast, responsive on desktop and mobile, and include a few basic examples showing how to use it in plain HTML and a couple of common frontend setups.