To claim full mediation is to claim that there is no direct effect. The tests for ADE test the null hypothesis that there is no direct effect, but failing to reject the null hypothesis is not the same as accepting the null hypothesis. Indeed, the confidence intervals for the ADE indicate that it is possible the direct effect is larger in magnitude than the indirect effect. The proportion mediated is a point estimate, but you can see that the confidence intervals include 100% mediated (as well as high values, i.e., that the direct and indirect effects are in opposite directions) and 50% mediated (i.e., that the direct and indirect effects are of equal size and direction).
You don't have to use a term like "full mediation" to describe the results in a meaningful and useful way. "We found evidence of mediation, but not of a direct effect." This explains the situation very clearly without using that specific term.
Going with the notation of VanderWeele and Vansteelandt (2009), with treatment indicator A, we can never observe $Y(A=1,M=0)$, because this is the outcome for treated individuals who are exposed to the mediator as if they were in the control group. Conversely, we cannot observe $Y(A=0,M=1)$. Therefore, these are quantities that require to be predicted in causal mediation analysis.
In summary, M(1) means M(A=1).
I add the following slide by Stijn Vansteelandt, which illustrates the notation quite nicely:
Edit: Some further clarifications as response to comments.
Sure you can have people also taking up the mediator in the control group, which is good, you can use that information to predict take up of the mediator under the counterfactual.
Let's say you are interested in the natural direct effect in an RCT. This effect is conditional on some baseline covariates C, which are sufficient to adjust for confounding of the mediator-outcome relationship (which is potentially confounded even if treatment assignment was random):
$\mathbb{E}(Y(A=1,M=0) - Y(A=0,M=0)|C)$.
Then you will need to "impute" counterfactual outcomes $Y(A=1,M=0 | C)$, which are unobserved. A common approach to do so is to use an auxiliary model on the way, in which you estimate the effects of $A$, $M$ and $C$ on $Y$. Using that model, you can predict the missing outcomes $Y(A=1,M=0 | C)$ and - given all structural and statistical assumptions hold - arrive at an unbiased estimate for the natural direct (and similarly the indirect) effect.
So those individuals taking up the mediator in the control group will give you valuable information for fitting this auxiliary model, making it easier (from a model-fitting perspective) to separate the effects of $A$ and $M$.
Best Answer
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.