You are right. If the book said that, it is wrong. I do wonder if it is a typo or a poorly phrased passage that lends itself to misunderstanding, though. As you show, $\exp(\beta_0 + \beta_1X)$ is the odds of 'success' predicted by the model. The odds ratio associated with a $1$-unit change in $X$ is $\exp(\beta)$. You can also think of is as the factor by which you would multiply the odds of 'success' associated with $x_i$ to get the odds associated with $(x_i + 1)$ (these are the same, phrased differently).
Since $p$ is the probability of success, $p/(1-p)$ is the odds of success by definition. Imagine $X=0$, then the right hand side simplifies to $\exp(\beta_0)$, but the left hand side is unchanged. Thus we can see that $\exp(\beta_0)$ is the odds of success when $X=0$. Now imagine we move to where $X=1$, then we can rewrite the RHS as $\exp(\beta_0)\times\exp(\beta_1)$ using the rules for exponentiation. Thus we can see that $\exp(\beta_1)$ is the factor by which you would multiply the odds of success to determine the odds of success associated with a $1$-unit increase in $X$.
I think the pdf you link to is sloppily written. At one point it says, "Odds Ratio. The odds ratio is equal to exp(B)", but there are no subscripts, whereas above that the intercept and the coefficient on X are differentiated by subscripts. That sets up the reader for a misunderstanding here.