Is there a difference between a causal relationship and a DIRECT causal relationship? The following site (http://www-ist.massey.ac.nz/dstirlin/CAST/CAST/Hcausal/causal_c2.html) defines a causal relationship as one where one variable 'directly' affects the other, but without the other variable having any influence on the first variable. However, the following site (http://medical-dictionary.thefreedictionary.com/direct+causal+association) defines a direct causal relationship as one where one variable causes a change in the other and there are no intervening variables.
However, both definitions are different. You can have a relationship with no intervening variables, but one where both variables directly affect each other. One wouldn't dare call this a correlation, but according to the first definition that wouldn't be a causation either.
Moreover, I assume that there are relationships where there are intervening variables but where one variable can cause a change in the other without the reverse being possible. According to the second definition, this wouldn't be a causation.
This begs the question: is there a difference between a causal relationship and a DIRECT causal relationship?
 A: Causal relationships can be direct or indirect. That is, direct causal relationships are a special case of causal relationships. For example, in $A \rightarrow B \rightarrow C$, the node $A$ is a cause of $C$ but it affects $C$ through $B$, so although there is a causal relationship between $A$ and $C$, this is an indirect causal relationship.  
About the relationship between causation and correlation, note that "correlation does not imply causation". For example, in $B \leftarrow A \rightarrow C$, $B$ and $C$ are correlated but do not have any causal relationship. 
A: 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.
