Correct way to present the definition a of Markov process of order $p$ for a vector process?

Usually when we define a Markov process of order $$p$$ for a univariate time-series $$\{X_t\in\mathbb{R},t=1,2,\cdots\}$$, the definition is presented as follows

$$$$P(X_t\leq x_t\mid x_1,\cdots,x_{t-1})=P(X_t\leq x_{t}\mid x_{t-p},\cdots,x_{t-1})$$$$ however, let us now consider a vector process $$\{Z_t=(x_t,y_t)'\in\mathbb{R}^2,t=1,2,\cdots\}$$. Would the correct way to present it be $$$$Z_t\mid Z_{1},\cdots,Z_{t-1}\sim Z_t\mid Z_{t-p},\cdots,Z_{t-1}$$$$ My intuition is that presenting it as we did for the univariate cse would be incorrect. However, I saw a paper where it had shown the process in that manner and now I am confused.