A. "Mediation" conceptually means causation (as Kenny quote indicates). Path models that treat a variable as a mediator thus mean to convey that some treatment is influencing an outcome variable through its effect on the mediator, variance in which in turn causes the outcome to vary. But modeling something as a "mediator" doesn't mean it really is a mediator--this is the causation issue. Your post & comment in response to Macro suggest that you have in mind a path analysis in which a variable is modeled as a mediator but isn't viewed as "causal"; I'm not quite seeing why, though. Are you positing that the relationship is spurious--that there is some 3rd variable that is causing both the "independent variable" and the "mediator"? And maybe that both the "independent variable" & the "mediator" in your analysis are in fact mediators of the 3rd variable's influence on the outcome variable? If so, then a reviewer (or any thoughtful person) will want to know what the 3rd variable is & what evidence you have that it is responsible for spurious relationships between what are in fact mediators. This will get you into issues posed by Macro's answer.
B. To extend Macro's post, this is a notorious thicket, overgrown with dogma and scholasticism. But here are some highlights:
Some people think that you can only "prove" mediation if you experimentally manipulate the mediator as well as the influence that is hypothesized to exert the causal effect. Accordingly, if you did an experiment that manipulated only the causal influence & observed that its impact on the outcome variable was mirrored by changes in the mediator, they'd so "nope! Not good enough!" Basically, though, they just don't think observational methods ever support causal inferences & unmanipulated mediators in experiments are just a special case for them.
Other people, who don't exclude causal inferences from observational studies out of hand, nevertheless believe that if you use really really really complicated statistical methods (including but not limited to structural equation models that compare the covariance matrix for the posited mediating relationship with those for various alternatives), you can effectively silence the critics I just mentioned. Basically this is Baron & Kenny, but on steroids. Empirically speaking, they haven't silenced them; logically, I don't see how they could.
Still others, most notably, Judea Pearl, say that the soundness of causal inferences in either experimental or observational studies can never be proven w/ statistics; the strength of the inference inheres in the validity of the design. Statistics only confirm the effect causal inference contemplates or depends on.
Some readings (all of which are good, not dogmatic or scholastic):
Last but by no means least, part of a cool exchange between Gelman & Pearl on causal inference in which mediation was central focus: http://andrewgelman.com/2007/07/identification/
An important first step of an analysis is define the estimand, or estimation target. When you write, "causal mediation effect", the estimand is left ambiguous. There are several possible terms you could be describing. I'll take you to mean the indirect natural effect.
For convenience, this answer will use counterfactual notation, with an asterisk denoting a counterfactual. With a single mediator, this is defined as $$\mathbb{E}[Y_{t, M_{1}^*}^* - Y_{t, M_{0}^*}^*],$$ for a possible treatment value $t$. It is interpreted as giving the causal effect of the treatment that flows through a mediator $M$ when the treatment is held at $t$ (to block the direct effect).
Your problem has multiple mediators, but their analysis is simplified because neither is a descendant of the other. Thus we can study the effect of each mediator on its own.
For the first mediator, the indirect natural effect is
\begin{align*}
& \mathbb{E}[Y_{t, M_{1,1}^*}^* - Y_{t, M_{1,0}^*}^*] \\
=\, & \mathbb{E} \left[ \{\beta_0 + \beta_1(\alpha + \alpha_1 + v) + \beta_2M_2 + \beta_3tM_2 + \beta_4t + \epsilon\} - \\ \{\beta_0 + \beta_1(\alpha + v) + \beta_2M_2 + \beta_3tM_2 + \beta_4t + \epsilon\} \right] \\
=\, & \beta_1\alpha_1,
\end{align*}
for any $t$. This is the value you described.
For the second mediator, the indirect natural effect is
\begin{align*}
& \mathbb{E}[Y_{t, M_{2,1}^*}^* - Y_{t, M_{2,0}^*}^*] \\
=\, & \mathbb{E} \left[ \{\beta_0 + \beta_1 M_1 + \beta_2(\gamma + \gamma_1 + e) + \beta_3t(\gamma + \gamma_1 + e)+ \beta_4t + \epsilon\} - \\ \{\beta_0 + \beta_1M_1 + \beta_2(\gamma + e) + \beta_3t(\gamma + e) + \beta_4t + \epsilon\} \right] \\
=\, & \gamma_1 ( \beta_2 + \beta_3 t)
\end{align*}
for any $t$. Thus the natural indirect effect depends on the treatment value $t$. Holding the treatment at $t=0$ or at $t=1$ will lead to a different effect of the second mediator $M_2$.
Each indirect effect can be consistently estimated by plugging in a consistent estimator of the regression coefficients.
Best Answer
I regret to say that kappa-squared has some pretty big flaws, and its continued use probably does more harm than good. I no longer recommend it. Please see:
Wen, Z., & Fan, X. (2015). Monotonicity of effect sizes: Questioning kappa-squared as mediation effect size measure. Psychological Methods, 20, 193-203.