Let $\{X_n\}$ be a Markov chain on a state-space $E$. A state $i$ is recurrent if $$P(X_n = i\;\text{for some} \;n\geq 1|X_0=i) = 1\tag{Definition 1}$$
$$\text{for some}\; n\geq 1\; P(X_n=i|X_0=i)=1\tag{Definition 2}$$
Definition $2$ is supposed to be incorrect. Could someone provide an example where definition $1$ holds but definition $2$ does not or vice versa? Or otherwise show me why they aren't equivalent. I've been told it might have something to do with when $n$ tends to infinity.
If $\{X_n\}$ is any Markov chain with finite state space such that $0<p_{ij} <1$ for all $i,j$ then the first condition holds because such a chain is (positive) recurrent. However condition 2) fails because all elements of the n-th power of the transition matrix are all also less than $1$. 2) implies 1) is obvious.