What are Markov chains?
A Continuous-time Markov Chain can be represented as a directed graph with constant non-negative edge weights.  An equivalent representation of the constant edge-weights of a directed graph with $N$ nodes is as an $N \times N$ matrix.  The Markov property (that the future states depend only on the current state) is implicit in the constant edge weights (or constant entries in the matrix).  Implicit means implied by.  Mathematicians use it as a euphemism meaning, "you should prove it yourself."

But the first paper defines a notation consistent with a Continuous-time Markov Chain, sometimes called a Markov Process, while the second paper defines a notation consistent with a Discrete-time Markov Chain.  They say


  $P$ is the transition probability matrix indicating the probability of getting from one state to another, and $\pi$ is the initial probability distribution representing the likelihood for the system to start in a certain state. [emphasis added]


They are assuming that the matrix is constant over time (thus implying the Markov property).  Implicit in the term probability is the fact that each constant is in range $[0,1]$, that the entries in every column of $P$ sum to $1$, and that the sum of the entries in $\pi$ sum to $1$.

I can't read the third paper, it is paywalled.  If the entries in every column of the matrix are required to sum to 1 then they are probabilities and they are talking about Discrete-time Markov Chains.  If the entries in every column can sum to an arbitrary number then the entries represent rates not probabilities and they are talking about Continuous-time Markov Chains.

Continuous-time Markov Chains are not the same as Discrete-time Markov Chains.  In a Continuous-time Markov Chain the edge weights do not represent probabilities, but rather transition rates.  The edge weights must be non-negative, but can be arbitrarily large, and the weights of the out-edges can sum to any non-negative number.  The sum is not required to be $1$.

With both Continous-time and Discrete-time Markov Chains the Markov property is implied by the constant edge weights (or equivalently the constant entries in the transition matrix.)