What is the precise definition of pseudo-polynomial time (feat. Counting Sort)
Let us denote the input's length by $n$, and lets remember that $n=\Theta\left(N\log K\right)$. You won't be able to write the exact running time of counting sort as an expression of the form $2^{\Omega(n)}$, as this would mean that for large enough inputs, the running time is always exponential. However, in this case, if $K$ is not too large compared to $N$, then the running time is actually linear, this means that there exists an infinite series of inputs with increasing size on which the running time is not exponential.

With this is mind, recall that what you seek is the worst case complexity of your algorithm, i.e. you want to find a certain time bound that holds for all inputs of large enough size. This means that to lower bound the complexity, it suffices to find one series of inputs with increasing size on which the running time is exponential. This allows you to focus on inputs where $K=2^N$, in this case the input size is $n=\Theta\left(\log^2 K\right)$, and the running time is $O(K)=O\left(2^{\sqrt{n}}\right)$, which is definitely exponential.

In general, a pseudopolynomial time algorithm is an algorithm which runs in polynomial time in the numeric value of the input (which means it is not necessarily polynomial in the input's length). Wiki's definition is no different, except it adds the restriction that the running time is polynomial in both the numeric value and the input size, i.e. if the input size is $n$, and its associated numeric value is $v$, we want a time bound of the form $p(n,v)$ for some polynomial $p$. This excludes for example a $2^N+K$ time algorithm for sorting $N$ numbers in the range $[1,K]$, although this is polynomial in $K$.