Coalescent Theory - Probability for $k$ alleles that of one coalescence event occured $t+1$ generations ago
Is this the exact text from the book? The left side seems to represent the probability for 


  "No coalescence in $k$ lines in $t$ generations (i.e. the $Pr(k)^t$ term), and at least one coalescence among those lines in generation $t+1$ (the $1-Pr(k)$ term)"


which is the same event as


  "First coalescence event in $k$ lines is exactly in generation $t+1$".


The right hand side is derived analogous to the second equation, with $1-\frac{1}{2N}$ ("no coalescence in two lines") replaced by $1-\frac{k\choose2}{2N}$ (approximation for "no coalescence in $k$ lines):

$$
\begin{align}
Pr(k)^t \left[ 1-Pr(k) \right] &amp;≈ \left( 1-\frac{k\choose2}{2N} \right)^t \left [1 - 1 +  \frac{k\choose2}{2N} \right] \\
&amp;\approx \exp\left( - \frac{k\choose2}{2N} t \right) \frac{k\choose2}{2N}
\end{align}
$$

were is first approximation is due to your first calculation. The second one seems to use a first order Tailor approximation of $\exp(x)$:  

$$
\exp(-xt) = exp(-x)^t \approx (1-x)^t
$$