One of the terms the teacher uses is self-correction. This is when the rider learns to correct themselves. It's essential because the horse needs training every day, and obviously the teacher can't be there watching at each session.
I think the concept of self-correction is a useful one for a writer. As I'm writing this blog post I'm constantly tweaking, choosing a different word, selecting a better verb for example or correcting my grammar, making it better, or clearer, or more effective all the time (I hope).
It's second nature to me - I've been a professional writer for over twenty years. If this were my creative writing, rather than a blog post, I'd be doing the same. And then I'd write a second draft with more self-correction, and then probably a third. At that point I show it to my outside readers.
But our first reader is ourselves, and the more we learn to self-correct the easier the transition process will be from idea to page to readable page.