Barron and Kenny assumptions are the following
- a change in levels of the exposure variable significantly affects the
changes in the mediator (i.e., Path from $A$ to $B$);
- there is a
significant relationship between the mediator and the outcome
(i.e., Path from $B$ to $C$);
- a change in levels of the exposure
variable significantly affects the changes in the outcome (i.e., total effect of $A$ on $C$ is significant);
- when the previously defined
paths are controlled, a previously significant relation between the
exposure and outcome is no longer significant, with the strongest
demonstration of mediation occurring when the path from the
independent variable to the outcome variable is zero.
Here $A$ is your treatment, $B$ is your mediator, and $C$ is the outcome.
what if A causes B and B causes C, but there is no direct effect between A and C?
Since there is no significance for the total effect of $A$ on $C$, assumption 3 fails.However, recent developments in science suggest that assumption 3 is not needed.
Consensus is that the relationship between $A$ and $C$ need not
be statistically significant for $B$ to be a mediator. The reason is that
the effect of $A$ on $C$ may not be significant when direct and
mediated effects have opposite sign.
It is obviously not mediation given that there is no direct effect of A on C (I would like to call this an indirect effect).
It is unclear why would you deviate from the traditional definition. Different definitions exist for different approaches, but all of them agree that indirect effect is associated with mediator.
Secondly, I read on another thread that if A affects B and B affects C, then mediation need not be tested, its assumed.
Again, this approach contradicts the assumptions 3 and 4. The scientific consensus is that those assumptions are too harsh and points have been made on how to relax those. I suggest reading the following for more up to date information about mediation(the last article contains an excellent summary of Barron and Kenny vs new approaches):
MacKinnon, D. P. (2008). Introduction to statistical mediation analysis.
New York, NY: Erlbaum.
Pearl, J. (2001). Direct and indirect effects. In J. Breese & D. Koller (Eds.),
Proceedings of the seventeenth conference on uncertainty in artificial
intelligence (pp. 411–420). San Francisco, CA: Morgan Kaufmann.
Valeri, L., & VanderWeele, T. J. (2013). Mediation analysis allowing for exposure–mediator interactions and causal interpretation: Theoretical assumptions and implementation with SAS and SPSS macros. Psychological Methods, 18(2), 137-150.
Also, here is a good discussion on whether Barron and Kenny's method is outdated.