Ok, forgive my ignorance, but I keep getting confused about something at the core of GLMs. Some textbooks describe the two main parts of a GLM as the link function and the distribution of the error terms. Others describe the two main parts as the link function and the variance function, where the variance function is a description of the relationship between the mean and the variance of the response (i.e., the response distribution). But the error distribution and the response distribution seem like different things to me.
If I had an equation where $Y_i = B_0 + X_i*B_1 + e$, I can see how for any given value of $X_i$ (plus constant values of $B_0$ and $B_1$), each random variable $Y_i$ would take on whatever distribution the error term had. But does that necessarily make the overall response distribution equal to that same error distribution?
Do my questions even make sense?