I’m an advocate of Ink, it’s 100% free and very flexible, and in my opinion has the best, non-intrusive, syntax. But it offers only the bare bones of a story engine. You won’t get a text input field out of the box as you wish, since that’s considered part of the “front end”.
ChoiceScript seems to be very close to what you’re looking for. But it’s proprietary. You can use it for free if you do not monetize your game, and that includes ads displayed in the same page as the game. If you want to make it commercial, you’ll either have to buy a license or contract with Choice of Games, which may or may not be interesting to you. Also, it doesn’t have a save system either.
Twine is the best after Ink, in my opinion, and a good compromise between Ink and ChoiceScript. You get a lot out of the box, and since it’s 100% free and popular, there’s a lot of resources out there. Tutorials, assets, tooling, plugins, what have you… I personally don’t like the macro syntax, though.
There are many other tools, but the less popular the harder it is to find good community support.