For me, the second draft isn't a revision, or a re-envisioning of the book I set out to write. The second draft is when I read through my first draft and add in details that I realize I left out in my rush to complete the first draft (the phase I like to think of as taking a roller coaster ride with my characters.) There's no time for pausing to consider the perfect word or the small, descriptive details that accompany the main action while you, as the writer, are riding the roller coaster. The second draft is the time to consider those.
The second draft is also the time when I look at what I've written and recognize the subtle threads that are woven through the story as a whole. These threads may only peek through here and there in the first draft, because they may be so subtle that I didn't recognize their significance while I was on the roller coaster. Now they spring out at me in vivid relief, and as I work on the second draft I can pull those threads through, so they find their proper place in the book as a whole.
The second draft is also the opportunity for adding in facts. I do a great deal of research before I get on that first draft roller coaster, but once I've taken off I always run up against questions that I discover I need to answer before the manuscript is ready to be revised. Interrupting the roller coaster ride in order to look up these facts can upset the flow of the writing,
so I leave notes or gaps in the first draft to remind me to find out about this or that, and the second draft is my chance to look up those facts and fit them in correctly. These can't be big facts that will alter the way your characters behave, or course, or how your plot turns out - those have to be answered before I climb on the roller coaster. But the second draft is my chance to research those little facts I didn't know I needed to discover when I first started writing.
Sometimes I don't even realize when my first draft turns into my second draft. This evening my husband asked how close I was to the end of the book. As I tried to explain that I'd already written the end, I realized what I've been doing these last couple of days was pulling threads through and adding in those descriptive details and those additional facts. I realized I've completed my first draft, and I'm partway through my second draft. That means my new book should be ready to give my critique group soon, and then read to revise and send to my agent. I've got to admit, I'm thrilled to realize I'm well into my second draft!