The two dichotomies are familiar: Atheism versus the broad sense of Theism, and Deism vs. the narrow sense of Theism. The reason I want to present the dichotomies this way is so that I can propose a new analogy: the pinball machine God.
Deists love their watchmaker God analogy. This lovely little world is created with awesome science, and the awesomeness of science proves that God is. Some deists might edge up the side of the side of the triangle a little.
On the other hand, narrow-definition theists might be offended at the puppeteer analogy. They'd also probably be offended that I classified Deism as having the same magnitude of involvement. But look at it this way: you construct and elaborate timing mechanism, flick the switch to place your bomb via an intelligent robot you designed, and then go out and get a cup of coffee, return a library book, and sit at a cafe overlooking a famous river of your choice. As you're hailing the waiter for the check, the bomb explodes a hundred miles off, destroying your evil arch-nemesis' secret lab.
OR, you fight a half dozen lackeys at the lab yourself, using your super-awesome martial arts moves, place the bomb by hand, light it with a match, and run out, just in time to feel the heat of the explosion on your back as you roll safely onto the grass.
Either way, you still blew up the place; it's just an issue of method.
And even though it doesn't really matter which method God uses, I'd now like to explain my idea of a pinball machine God, which sits pretty close to the center of the triangle. God constructs this elaborate machine for us: the pinball machine we call Earth. There are an uncountably many number of targets, bumpers, balls, and flippers. Maybe we're the balls, but God is certainly the player. The coin is inserted, and God mutters under her breath: let there be light. And the machine comes to life.
See, the the pinball machine world, there's a lot of factors. Every ball starts with a unique trajectory, maybe some special dents and scuffs too, or perhaps they pick them up on the way. Maybe they have different masses, radii, and densities. They bounce around making and missing targets, ricocheting off of bumpers and running into each other. An then, every once in a while: fwip! They're hit by a flipper. Maybe some balls are flipped all the time, and maybe some balls are basically never flipped.
The core of the analogy is this: it may be that God constructs the world and influences it certain ways, but that there's a good amount of randomness inherent in the system. Random here doesn't mean that God doesn't know about or account for problems or peculiarities, but that God can't do too much about it anymore because that's intentionally the way he built the game. Part of the joy in creation, I'd imagine, is watching something flourish on it's own. Flourish? Okay, maybe God the Gardener would have been a better analogy...