Landing page or directly show the product/demo?

I don't think it is that important, what you call it as long as there is a demo where users can quickly understand what exactly is this about. I also agree that making people sign up before they can even try their own prompt is a no-no. Let them try maybe five prompts for free and then force the sign up.

Oftentimes, Landing pages are just waitlists, with no idea on when the product will be available or what the pricing will be. While they can help makers assess product - market fit, it can often be frustrating for those who signed up. If the product is available as a minimum viable product or a beta, landing pages are specific use cases for specific audiences and campaigns, leading to direct signup to the product. For example, my product is ready and I am doing two campaigns, one to say product managers of large development houses and another to indie makers. The use cases for both these audiences are different. So, I create two different landing pages, highlighting the specific use cases and run two different campaigns that lead the audiences to the respective landing pages.

Why not both? A demo to show how the product works but also some additional info as a landing page. Though looking at your current page, I think the site would be very annoying for users, since they'll type something in and then be met with "please sign in to use". I know GPT-3 costs money to run and you want to limit people's usage with accounts, however, I would recommend you don't have a demo on the page in that case since it's a little deceptive. Only have the demo show up once they create an account and just outright say "play with our demo once you create an account" You can have a video showing how it works otherwise for those that have not logged in.

Landing page - Helps reduce cognitive load and smoothenes the journey of adoption