On the definition of Markov chains A Markov chain with discrete time dependence and stationary transition probabilities is defined as follows. Let $S$ be a countable set, $p_{ij}$ be a nonnegative number for each $i,j\in S$ and assume that these numbers satisfy $\sum_{j\in S}p_{ij}=1$ for each $i$. A sequence $\{X_j\}$ of random variables with ranges in $S$ is defined to be a Markov chain if
$$
P[X_{n+1}=j|X_0=i_0,\ldots,X_n=i_n]=p_{i_nj}\quad(1)
$$
for every $n$ and every sequence $i_0,\ldots,i_n$ in $S$ for which 
$P[X_0=i_0,\ldots,X_n=i_n]>0$.
The property
$$
P[X_{n+1}=j|X_n=i]=p_{ij}\quad(2),
$$
is a consequence of the definition (1), but I don't think that (2) implies (1). So are there any interesting examples which satisfy (2) but are not Markov chains?
(Of course (1) determines $P[X_{n}=i_n,X_{n-1}=i_{n-1},\ldots,X_{n-k}=i_{n-k}]$ in terms of the transition probabilities and the initial probabilities for any $n$ and all $k\leq n$, whereas (2) does so for any $n$ and for $k=0,1$.)

Any probability distribution at all will satisfy your equation 2. The point of a Markov chain is that in the equation 
$$
P[X_{n+1}=j|X_0=i_0,\ldots,X_n=i_n]=p_{i_nj},
$$
the probability only depends on the previous element:
$$
P[X_{n+1}=j|X_0=i_0,\ldots,X_n=i_n]=P[X_{n+1}=j|X_n=i_n].
$$
