The Wikipedia article of a diffusion process mentions:
In probability theory and statistics, a diffusion process is a solution to a stochastic differential equation. It is a continuous-time Markov process with almost surely continuous sample paths.
Are these two definitions equivalent?
The second definition is the correct one .
Using the same notation and hypothesis as in the Wikipedia's article of SDE, the strong solution of $$dX_t = \mu( X_t) dt+ \sigma (X_t) dB_t, \ X_0 =Z $$
is a continuous-time Markov process with almost surely continuous sample paths (this kind of diffusions are called Itô diffusion, in some cases you should deduce by context if they are referring to this kind of process).
However not every kind of SDE defines a Markov Process (here is a nice example: Proof that time integral of OU process is not Markov while together with the integrand it is. ).
On the other hand, not every Strong Markov processs with continuous sample paths solve an SDE (see for example: Do there exist diffusions that do not solve any SDE? ).