I am trying to explain the differences between the following concepts to someone and I realized I myself am super confused:

Continuous/discrete Markov Process

Continuous/Discrete Markov chains

Markov property : $P\{X_n=i|X_{n-1}=j,X_{n-2}=k,...\}=P\{X_n=i|X_{n-1}=j\}$ ?

I used to think: Every process that has Markov property is a Markov Process. Every Markov process is a Markov chain and every Markov chain is a Markov process.

But it seems crazy now when I think about it, because if they are all the same, why there are different names for them?

And they are continuous (discrete) if their parameter set $T$ is continuous (discrete) regardless of their state space?

I want to start with homogeneous Markov chain and process too. But since I am already too confused and Wikipedia is making me more confused, I prefer to wait till I get these basic definitions straight first.

Also if anyone has any suggestions on how to explain these terms w/o causing any confusion I would appreciate it a lot. (I would also contact my SP teacher to teach him that, coz he has clearly taught us bad)

Thanks a lot