Understanding the Difference Between Different Types of Markov Chains? I have been trying to learn more about different types of Markov Chains. So far, here is my basic understanding of them:
Discrete Time Markov Chain: Characterized by a constant transition probability matrix "P"
Continuous Time Markov Chain: Characterized by a time dependent transition probability matrix "P(t)" and a constant infinitesimal generator matrix "Q". The Continuous Time Markov Chain is based on the Exponential Distribution (i.e. sojourn times/holding times must be Exponentially Distributed) and thereby must obey the Memoryless Property.
Non Homogenous Continuous Time Markov Chain: Characterized by a time dependent transition probability matrix "P(t)" and a time dependent infinitesimal generator matrix "Q(t)". Non Homogenous Continuous Time Markov Chains are not necessarily based on the Exponential Distribution and thereby do not need to obey the Memoryless Property.
This brings me to my question - given the above information, I am having difficulty understanding the difference between these Markov Chains and a Semi-Markov Process.
Based on some readings that I have done, it seems like a Semi Markov Process is characterized by its "Sojourn Time" - that is, the amount of time that is spent in some state. Since the probability distribution of this "Sojourn Time" does not necessarily need to be Exponentially Distributed, a Semi Markov Process does not need to obey the Memoryless Property. If this is the case, then how exactly is the Semi-Markov Process different from a Non Homogenous Continuous Time Markov Chain?
Can someone please help me understand this? Perhaps some example could illustrate in which conditions it might be more advantageous to use a Semi-Markov Process compared to a Continuous Markov Process and vice versa?
Note: I am currently reading this reference : https://math.stackexchange.com/questions/2960929/difference-between-non-homogeneous-markov-and-semi-markov

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*Note 1: I understand that as the name suggests, "Semi-Markov" does not need to obey the "Markov Property". That is, in Semi-Markov, the future states that the process can go to does not only depend on the current state, and can also depend on the history of the process. This is in contrast to a Continuous Markov Process in which the Markov Property must be obeyed, and the future state of a Continuous Markov Process can only depend on the current state.


*Note 2: I am told that one of the differences between Semi-Markov and Markov is how the notion of time is recorded. In a Continuous Markov Process, time is recorded in the "clock forward" format - this means that all time transitions are recorded from some initial time. In a Semi Markov Process, time is recorded in the "clock reset" format - this means that the clock is restarted every time a transition is recorded. Supposedly, "clock forward" is not able to take into consideration the history of the Markov Process - whereas "clock reset" is able to take into consideration the history of the Markov Process. However, I do not understand this point - I am not sure as to why recording time using different formats "magically" allows the Markov Process to consider the history or not to consider the history of the process.


*Note 3: It seems to me like a Semi Markov Process is closer to a "Discrete Markov Chain" : If my understanding is correct, a Semi Markov Process does not have a P(t) matrix or any type of Q matrix - a Semi Markov Process only has a time independent P matrix. Is this correct?


*Note 4: Again, if my understanding is correct, it seems as though a Semi Markov Process might not be as useful for modelling real world phenomena compared to a Continuous Markov Process (i.e. when modelling a person transitioning between medical condition, it will likely be useful to know the history of how long he has been healthy or sick) - with this being said, in what kinds of situations is it useful to use Semi Markov Processes?

Let's rewind the definitions.
Definition $1.$ Let $E$ be a Polish space. Consider $\langle X_t\rangle_{t\in\mathcal I\subset \mathbb R}$ an  $E$-valued stochastic process. It follows the Markov property if $$\mathbf P[X_t\in A\mid \mathcal F_s] = \mathbf P[X_t\in A\mid X_s,]~~~~\forall A\in \mathfrak B(E),~\forall s,t\in \mathcal I\wedge (s\leq t) \tag 1\label 1.$$
Any process following $\eqref 1$ is a Markov Process. With $E$ being countable and $\mathcal I = \mathbb N_0,$ the process becomes discrete Markov chain.
$\bullet$ Does the specification of homogeneous and non-homogeneous change the defining property of the Markov process, viz. $\eqref 1?$
Defintion $2.$ $\langle X_n, T_n \rangle_{n\in \mathbb N}$ is a Markov renewal process if $$\mathbf P[ X_{n+1}=j, ~T_{n+1}- T_n\leq t\mid X_0,\ldots, X_n; T_0, \ldots,T_n] =   \mathbf P[X_{n+1}=j,~ T_{n+1}- T_n\leq t\mid X_n]\tag 2\label 2.$$
It is assumed that $\langle \mathbf X, T\rangle$ is time-homogeneous i that $$\mathbf P[X_{n+1}=j,~ T_{n+1}- T_n\leq t\mid X_n] := Q(i, j, t)\tag 3$$ being independent of $n.$
Definition $3.$ Define $Y_t := X_n\cdot\mathbb I_{[T_n,~T_{n+1})}(t).$ Then $\langle Y_t\rangle$ is the semi-Markov process associated with $\langle X, T\rangle.$
$\bullet$ Markov property holds for $Y_t$ only at epochs $T_n.$
$\bullet$ The distribution of the sojourn time $T_{ij}$ can be anything.
$\bullet$ If $Q(i,j, t) = p_{ij}[1-e^{-\lambda(i)t}],$ then only the semi-Markov process becomes a time-homogeneous Markov process. $\langle Y_t\rangle$ is a Markov process.
Therefore, a Markov process has the Markov property at all times. A semi-Markov process has the Markov property only at epochs.
Addition of non-homogeneity doesn't make a drastic distinction. As aptly remarked here:

In effect a semi Markov process can have whatever transition probability dependence on time you like, whereas in a non-homogeneous Markov chain you will still always have the probability of a transition from $i$ to $j$ in $[t,t+h]$ being $q_{ij}(t)h+o(h). $

At the end of the day, it is always the case of whether the Markov property $\eqref 1$ has been followed throughout or only at some instances.

References:
$\rm [I]$ Introduction to Stochastic Processes, Erhan Çinlar, Dover Publications, $2013,$ chapter $10,$ p. $314.$
$\rm [II]$ Stochastic Processes, J. Medhi, New Academic Science Limited, $2012,$ chapter $7,$ p. $299.$
