As far as I understand it a stochastic process is a mathematically defined concept as a collection of random variables which describe outcomes of repeated events while a deterministic process is something which can be described by a set of deterministic laws. Is then playing (classical, not quantum) dices a stochastic or deterministic process? It needs random variables to be described, but it is also inherently governed by classical deterministic laws. Or can we say that throwing dices is a deterministic process which becomes a stochastic process once we use random variables to predict their outcome? It seems to me only a descriptive switch, not an ontological one. Can someone tell me how to discriminate better between the two notions?
Physics models rarely hint at ontological level. Throwing dice can be modelled as deterministic process, using initial conditions and equations of motion. Or it can be modelled as stochastic process, using assumptions about probability. Both are appropriate in different contexts. There is no proof of "the real" model.
Throwing dice is just throwing dice. That's all. It's not stochastic, nor deterministic. It's just throwing dice.
Now we model throwing dice as a process, and that's where the stochastic or deterministic side starts to play in. It is the process that is stochastic or deterministic, not the throwing of the dice. It's how we think about the throwing of dice that can be stochastic or deterministic.
When science really pushes, the model it sees of the world is a bit of both. The models which are most popular in science are models where the rules for the time evolution of systems are deterministic, but the initial conditions are not perfectly known. For example, you do not know the precise state of repair of your actin and myoisn in your muscles, and that will give you an unpredictable result as you throw the dice. This is most strongly evident in the field of statistical mechanics, where one models the inputs to a system with random variables and explore how they transform as time evolves. Note here the word "model." The inputs do not need to be random... they merely need to be convenient to be modeled as-if they were random.
At some point, we choose to simplify. Rather than claiming we don't know enough about the initial state of our arm muscles, we start to claim that the process itself is stochastic. We start to claim that "If I threw the die exactly the same, it lands on a different side." This is not accurate at the deepest understand that physics has to offer (you merely don't know enough not have the control to succeed at throwing it exactly the same), but it turns out to be good enough to make sense of the world around us -- and our games of chance. We start to claim that "throwing dice is a stochastic process." We may know that, at the deeper levels, the process is deterministic, we find it reasonable to model the throwing of dice as a way to turn a known input (the orientation of the die in your hand) into a random output (the orientation of the die at the end of the process).
It's still just throwing dice. That's all it is. But our model changed, and with our model, we changed whether we think of it as a stochastic or deterministic process.
Personally I find the dividing line between a stochastic process and a deterministic process is often whether one feels that someone could know enough information to predict the result. For example, by the Copenhagen Interpretation of QM, it is believed that it is simply not possible to know some combinations of things due to uncertainty. It is believed the universe simply does not let us see them. On the other hand, one can look at the brilliant work of Richard Turner, a card mechanic who makes a living by demonstrating that he can indeed know things about decks of cards that you might otherwise think were properly shuffled -- you are lead to believe you are witnessing a stochastic process, when it is so terribly deterministic in his hands!
Look up Diaconis's work on flipping coins. While it is technically deterministic, what happens is that extremely small changes in the initial conditions flip the outcome. The same would be true of dice. When you shake them in your hand and throw, small changes would give different outcomes. What makes it seem random is that we can't control our hands well enough to reproduce exactly the same throw (although some people are able to throw dice without making them tumble).
As you commented, rolling classical dice is a deterministic chaotic process, and thus impossible to model in enough detail to predict which face will be up, unless you have very accurate input data.
More accurate than is plausible for dice rolled by hand, unless maybe you're collecting it with high rez / high-speed cameras or other 3D motion capture system, and have carefully measured the physical properties of the dice and the surface.
A simpler model of the same thing is that it's just random which way up they land.
There might be some scope for predicting which way the face is rotated, and significantly more scope for predicting where the die lands / dice land on a flat table, especially if the table is soft so the dice don't bounce a lot.
It depends on the amount of information you have about your coin flips, and the accuracy of your physical laws. If you have perfect information about every single particle comprising the coins, every air molecule in the room, every molecule in the coin-flipper (including neurons, nerves, muscles), every force acting on everything, and you have perfect physical laws to apply and a computer powerful enough to process everything, then it's deterministic. If you don't have that, it might as well be random.