Counterfactuals

(I get the idea that, generally speaking, the layperson tends to believe philosophy as an active practice is dead. Philosophy, they believe, is merely the study of the history of thought. And it certainly is. A huge chunk of philosophy is interpretation and study of what other great thinkers thought. But there is actual philosophizing going on today, believe it or not.)

My big project for late this summer and into the fall semester has to do with counterfactuals. What is a counterfactual? Well, let's review some related concepts:
A counterfactual is a type of conditional, of which there are many.
First there is the 'Indicative Conditional':
"If the moon is made out of cheese, then Sally would like some cheese." What makes an indicative conditional true? Well, it is certainly controversial. The common-sense interpretation is that the antecedent (what comes after 'if') must cause the consequent (what comes after the 'then'). When conditionals are interpreted in this way, they are called 'causal conditionals'.
The problem with causal conditionals is that 'cause' is an extremely mysterious concept in any philosophical conversation, and a lot of people think it invokes implicit counterfactuals. This is a problem, as we shall see later.
So for simplicity's sake, most basic, non-modal systems of logic interpret "If the moon is made out of cheese, then Sally would like some cheese," as being logically equivalent to "Either the moon is NOT made out of cheese, or Sally would like some cheese." Logical equivalence means that the truth-value of the first statement must always be the same as the truth-value of the second statement.
Ok, think about this--it takes a moment to grasp.
Now this way of interpreting conditionals works a lot--the only counterintuitive result is that any conditional with a false antecedent is automatically true as a whole. For example-- "If the moon is made of cheese, 2+2=74" is true, because it is logically equivalent to "Either the moon is NOT made of cheese, or 2+2=74," which is true.
There is another way of interpreting conditionals, called 'strict conditional'. This interpretation attempted to alleviate the problem just mentioned by saying only a conditional with a true antecedent AND a true consequent could be true.

Alright, finally to counterfactual conditionals, or subjunctive conditionals (they both refer to the same thing from different literature).
When someone uses a counterfactual conditional, they realize that the antecedent is false, but are trying to say what would be the case if it weren't. For example:
"If Bush had been a veteran, his approval ratings would be higher."
This statement makes sense to all of us, regardless of whether we agree with it or not. But it isn't trivially true or trivially false. So here is the problem--how can we declare its truth-value? We can certainly try to show evidence for or against it--but what evidence is even relevant?
The Bush/veteran case seems to be rationally resolvable, but it seems there are much more difficult cases, such as "If George Bush had been named 'Jeb', the president of the United States in 2003 would not have invaded Iraq." What makes this counterfactual true? If George was named Jeb, it seems our world would be different in ways we can't imagine. Is the speaker saying someone named 'Jeb' wouldn't have invaded Iraq? Or are they saying a 'Jeb' couldn't have won, and Gore wouldn't have invaded? Or Gore would have prevented Sept. 11? There are so many circumstances which we are ignorant of, it isn't clear how the truth value can be determined.
Its also entirely plausible that we just can't talk about situations that bizarre and expect to come up with a decent answer.

But the point and the question in all of this that I am trying to answer is as follows: What is the formal (logical) definition of a counterfactual? Do we interpret them like causal conditionals? Do we consider a world in which all facts are the same besides the explicitly stated antecedent?

This is all very confusing, and I probably haven't explained it well. For that I apologize.