Neither causation can say they are correlated, nor correlation can say they are causation. Am I right ? Suppose there is no correlation between A and B. Does it mean there is no causal relation between A and B? No. It could be the case that A causes B, but C causes not-B. (This is modelled as defeasibility in causal logic.) And if that is the case, the combined presence of A and C in a given situation would neutralize each other's causal effects, with the result there would be no observed correlation.
No. Correlation is a prerequisite for causation. So if you know that A is the cause of B you also know that there is a "co"-"relation" between A and B. A is at least partially the cause of B and B is at least partially caused by A.
So it would be somewhat weird to know that A is the cause of B and not know that the two are related to each other.
In other words you wouldn't find yourself in a position where you know there's a causation, but being in search of a correlation. It's usually the other way around once you find a correlation you can take that as a hint of a causal relationship as a correlation is a prerequisite, but not a sufficient condition, for a causation.
What you describe is a completely different scenario of a superposition of different effects, which makes it difficult to test for correlate and causation. So you'd usually set up "lab conditions" where you test for just one parameter, either by suppressing the rest or by keeping them the same except for one.
You are right in saying that correlation is distinct from causation. Two or more sets of events can be perfectly correlated without one set being the cause of the others. For example, if I have a garden sprinkler with ten jets, I can vary the pressure of water through the sprinkler by opening and closing a tap. All of the jets will react to an opening tap by becoming more forceful, and to a closing tap by becoming less forceful. The waxing and waning of each jet will be correlated with the waxing and waning of all of the others, but there is no causal relationship.
It is also possible for there to be causation without any readily identifiable correlation, particularly where causes are essentially fuzzy. Take the statement that market sentiments determine stock valuations- it would be hard to deny the truth of the claim, but equally hard to figure out any meaningful correlation because the idea of 'sentiment' is inherently vague.
Causation is the relationship "X if and only if (Y and 'Y is in the past of X')".
Correlation is the trivial relationship "X and Y".
You can plug any combination of stuff into X and Y here. In your example you've plugged B into X, and "A and not-C" into Y, so the causation is that "A and not-C" causes B. A is not "A and not-C", A is A.
A is the sole cause of B
C is the sole cause of not-B
AC = A & C
P(B|A) = 1
P(~B|C) = 1
P(B|AC) = ? *
P(~B|AC) = ? *
In digital circuit design when signals try to drive a flip flop to both 1 and 0 (representing B and ~B) at the same time this causes an uncertain outcome or possible race condition. Applying that intuition if A is the sole cause of B, and if C is the sole cause of not-B, then the operative joint cause AC = A & C would produce an indefinite probability for the observation of B or ~B. The * is provided because I have not reviewed the math for applying Bayes' rule with two conditions that drive a contradictory outcome. It may be there is a definite probability for each outcome in the math model.
P(B|A) = 1
P(~B|C) = 1
Correlation is not causation so the mere fact of correlation does not imply that A is the sole cause of B or that C is the sole cause of ~B.
There is indeed an awkward disconnect, as far as I can see, between what is said in the context of statistics (correlation is not causation) and what is said in the context of philosophy of science, where correlation is at the heart of causation. Of course, Humean causation involved more than correlation - cause must be spatially contiguous with effect, cause must precede effect in time, and all similar events must be followed by an similar effects. (Details may vary). Nonetheless, the disagreement between the two is striking.
Mill's four methods of induction are helpful to further distinguish "mere" correlation from causation and high-light the importance of experimentation. See J.S. Mill - Stanford Encyclopedia of Philosophy especially section 3.3 (The italics are mine.)
“We may discover, by mere observation without experiment, a real uniformity in nature” (System, VII: 386). Where possible, however, engaging in experimental activity is the most direct way to uncover the causal relations between events, because it allows us to “meet with some of the antecedents apart from the rest, and observe what follows from them; or some of the consequents, and observe by what they are preceded” (System, VII: 381). Mill claims that, as science has progressed, four methods have emerged as successful in isolating causes of observed phenomena (System, VII: 388–406; see Cobb 2017: 240–1). Firstly, the Method of Agreement: where instances of phenomenon A are always accompanied with phenomenon a, even when other circumstances accompanying A are varied, we have reason to believe that A and a are causally related. Secondly, the Method of Difference: where the only distinguishing feature marking situations in which phenomenon a occurs or does not occur is the presence or absence of phenomenon A, there is reason to think that A is an indispensable part of the cause of a. (If we have noted, via the Method of Agreement, that in all instances of A, a is present, we can, where possible, systematically withdraw A, to determine whether A is a cause of a by the Method of Difference. Mill terms this the Joint Method of Agreement and Difference.) Thirdly, the Method of Residues: against the knowledge that A is the cause of a, and B the cause of b, where ABC causes abc, and AB causes merely ab, we can (by ruling out that c is the joint effect of AB) regard C as the cause of c. Fourthly, the Method of Concomitant Variations: whenever a varies when A varies in some particular manner, a may be thought to be causally connected to A.
However, that article also refers to another possible differentiation:-
A detailed anthropological study of the history of successful scientific practice is likely to reveal the irreducible use of imaginative hypothesis-making—not to mention changing questions and ideals of the sort later highlighted by Thomas Kuhn (1962). Such was the basis for a telling historico-normative debate between Whewell and Mill—the former arguing that scientific reasoning had and should involve the creative a priori development of concepts prior to the discovery of laws, the latter claiming, as can be seen in the quote above, that observation and induction alone could track facts about the world and elicit the concepts used in science.