Confusion with analysis of hashing with chaining
The reason that we are using $\Theta(1+\alpha)$ rather than $\Theta(\alpha)$ is that $\alpha$ could be very small. Imagine for example that the hidden constant in both $\Theta$s is one, and that $\alpha=0.000001$. There is a large difference between $1.000001$ and $0.000001$.

Regarding your wrong deduction, note that the first paragraph mentions average case and the second worst case. It's true that the average case is at most the worst case, but not the other way around.