Unobservable is any factor that is correlated with outcome and switch, which you do not model or do not have data for.

To identify a causal effect of x on y, you need to exclude the possibility that x is correlated with z and z influences y. If it is so, you need to model z in some way (here, it depends on what we talk about). There are also other considerations, but that is perhaps the most pressing and obvious one. You can quickly convince yourself that the above must hold if a causal effect is to be identified. Under assumptions, for example induced by experimental conditions, the above is also sufficient to identify a causal effect.

The absence of any unmodeled z ( eg the Bond villain in our aforementioned randomness experiment) is essentially always an assumption and not a provable fact, which motivates my earlier discussion. Note also that measuring, modeling or otherwise taking into account any such Z depends on the Z and therefore the data generating process (aka the situation at hand )

Given that causal identification requires assumptions on the DGP, no general or non model based definition can exist.
That is, we CAN say what we mean by causality, however the task of identifying causality has to be stated in terms of a of the data generating process.

Perhaps herein lies the confusion.

Based on the DGP, mathematical definitions of causality do of course exist. For example, in the framework of switch and outcome, as defined, causality is semiparametrically identified if the switching is uncorrelated with other factors that are not modeled. In that case, we identify the qualitative direction of causality, or causal effect. For example, if the switching is true randomness, as per your definition, then the condition is fulfilled. However, randomness is only sufficient not necessary.

What a causal effect is, well, that means depends on the data and situation, for example what values outcome can take and what measures you apply: Expectations, Distributions, Order Statistics and so on.
If we see three people operating switches in terms of dials, we would for example be able to identify the average marginal effect of dialing on outcome. This would be an estimated expectation.

Formal definitions of these concepts can be found in, hopefully, any modern statistics textbook.

Otherwise, there are many formal treatments available. For the concept of statistical identification, Lewbells article „Identification Zoo“ is a good introduction. Formal definitions of identification and causal identification (albeit written for econometricians) are presented.
http://fmwww.bc.edu/EC-P/wp957.pdf

The first sections give a good understanding why it only makes sense to talk about causal effects and causal identification, if we have a model (restrictions on the DGP) in mind.
While the text is generally about statistical identification, you can do the following: Assume that based on your DGP/model, theta_0 (in the text) is the parameter telling you, whether X causes Y. Then, reading section 3 is probably sufficient to get an understanding of what causal identification is.

As it applies to causality and counterfactuals specifically, read Pearl as one framework, and research on Rubin for another. There are now, I think, good books on amazon treating both in more generality, especially outside of econometrics (which is or was for some time leading in non experimental causal analysis). Then, literature on experiments is also concerned with causal analysis, though since the distinction between correlation and causation is less of an issue, the literature has less focus on stating what exactly makes a procedure a causal analysis and what does not.

For that reason I actually recommend looking at the non experimental literature first, since the need for definition arises.