Both sources you link to are pretty bad.
The clearest approach to causality is the one using structural equations, potential outcomes, and causal graphs [1]. In that appraoch:
Causal effects are assumed to exist even if one may not be able to determine them.
This is just logical: One needs to define causality first before one can even think about identifying it from data.
Causal effects are defined as the result of minimal external interventions to the value of a variable. E.g., the causal effect of variable A on variable B in unit $i$ is defined as $B^{a}_{i} - B^{a'}_{i}$, where $a, a'$ are two different fixed values. If for at least one unit and two different $a, a'$, these hypothetical values of B differ, A is said to have an effect on B.
Direct causal effects and what you might call "total" causal effects are both causal, but potentially different. The total causal effect is what I just described. Regarding direct causal effects, these are usually defined with respect to intervening/mediating variables, and there are actually two distinct types of them. One of them is the controlled direct effect. This could be $B^{a, m}_{i} - B^{a', m}_{i}$: One compares two hypothetical outcomes for different values $a, a'$, but the same fixed value $m$ for the mediator M (mediator as in $A \rightarrow M \rightarrow B$). The natural direct effect is defined as $B^{a, {m^{a}_{i}}}_{i} - B^{a', {m^{a}_{i}}}_{i}$: A still switches from a to a', but M is fixed at its hypothetical value under the intervention $A = a$.
- Obviously, researchers might know that a total causal effect exists and what its magnitude is, but they might not have a single clue about the mechanism/direct effects of a variable
- There may be causal loops, e.g. A influencing B and B influencing A, although these (usually?) cannot occur simultaneously, so the "loop" is an approximation that neglects the time lag by which B influences A back and so and so forth. Classic example: The mutual impact of prices and quantities of a product in an economy.
- Although it is unlikely and often ruled out by assumption, there may be causation without correlation (or rather: without dependence). This can be, for example, when confounding exactly cancels out the dependence induced by causation.
To expand on the last point, you may you measure $P(Y|D = 1) - P(Y|D = 0)$ as the dependence between treatment D and outcome Y. By fundamental properties of counterfactuals and some algebra, this is
$P(Y^{1}|D = 1) - P(Y^{0}|D = 0) = $
$P(Y^{1}) - P(Y^{0}) + (P(Y^{0}|D = 1) - P(Y^{1}|D = 0))$
where $P(Y^{1}) - P(Y^{0})$ is the true causal effect and $P(Y^{0}|D = 1) - P(Y^{1}|D = 0)$ is a bias term due to confounding. These two terms might exactly cancel out, so that $P(Y|D = 1) - P(Y|D = 0)$ = 0 even though $P(Y^{1}) - P(Y^{0}) \neq 0$. Again, this might be unlikely if the system you study is complex, but it may happen in some circumstances, regardless of how much data you have.
[1] Pearl, Judea 2009. Causality. Cambridge University Press.
