An inherent inconsistency in interpreting the indirect effect in a mediation analysis? 
Disclaimer: I am well aware that correlation and causation are different things, yet I will here write "A causes B" to keep it short.
This is in any case completely unrelated to the point I try to make.

Assume that we are interested in the effect of variable $X$ on variable $Y$, through a possible mediating effect of $M$ (all variables are assumed continuous here for simplicity).
The indirect effect of $X$ on $Y$ is defined as the product of $a$, the slope in the regression $M \sim 1+Y$, and $b$, the slope of $M$ in the regression $Y\sim 1+X+M$.
Now one would like to naively interpret this prodct $ab$ as follows: a change of $X$ by $+1$ induces a change of $M$ by $+a$, which in turn induces a change of $Y$ by $+ a\times b$.
However, this interpretation is wrong, because under classical assumptions (especially no interaction between $X$ and $M$ with respect to the DV $Y$, and a linear dependence of $\Bbb E (Y)$ on the predictors), the slope $b$ is the change in $Y$ when $M$ increases by $1$ and $X$ is kept constant. It is an "average simple effect", or simply the simple effect when there is no interaction, of $M$ on $Y$ controlling for $X$.

So on the one hand we have

*

*$X$ increasing by $1$ causes $M$ to increase by $a$ which causes $Y$ to increase by $a\times b$
and on the other hand we have

*

*the claim that increasing $M$ by $a$ induces an increase of $Y$ by $a\times b$ requires keeping $X$ constant.


This is contradictory and goes to show that the naive interpretation of  the indirect effect $a\times b$ is wrong.
What would be a more satisfactory interpretation of it then?

There seems to be a confusion that you can move variables only one at a time. You have to look at the whole pictures, not just part of it. There seems to also be a confusion on regression coefficients being on the "same level".
Take a linear model like, $y \sim\beta_1x_1+\beta_2x_2$, you can predict $y$ if you induce a $+1$ changes in both $x_1$ and $x_2$ even though both are estimated while both are held constant. It is $\beta_1 + \beta_2$.
In mediation analysis, you have to look at the whole picture. Assume the model :
$m \sim \alpha x$
$y \sim \gamma x + \beta m$
There is a total effect of $x$ and $y$, which we will call $\lambda$. In mediation analysis, the total effect is divided in a direct effet $\gamma$ of $x$ on $y$ and an indirect effect $\alpha \beta$, which are related as $\lambda = \alpha\beta + \gamma$.
If you increase $x$ by $+1$ regardless of $m$, then $y$ is increased by $\lambda$. In the mediation model, we explain that the total effect $\lambda$ had been divided in the direct effect $\gamma$ and indirect effect $\alpha \beta$. So, if you increase $x$ by $+1$, then there is the direct effect on $y$ from $x$ which is $\gamma$, and another effect on $m$ which is $\alpha$. If you keep $m$ constant, then the indirect effect is the product of $\alpha$ times the effect of $m$ on $y$, $\beta$, which is indeed $\alpha \beta$.
Now, if $m$ is increased by $+1$ while keeping $x$ constant, there is no indirect effect $x$; $y$ is increased by $\beta$. However, if you increase $m$ by $+1$ regardless of $x$, it becomes obvious that $y$ is increased by $\beta(1-\alpha^2)+\alpha\gamma$ instead, which the total effect of $m$ on $y$. This total effect is attributed to the effects of $x$ and $m$ on $y$.
In fact, "no interaction between X and M with respect to the DV Y" is the very fact why the assumptions are relevant, because interpretation would not be so straightforward.
