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.
Basically, when you add variables to a regression, no matter what those variables are, the coefficient on exposure can increase or decrease. You generally cannot learn anything about the causal structure of your system by observing coefficient change when adding or removing covariates to a model.
Often, adding a mediator will decrease the coefficient on the exposure because part of the effect of the exposure on the outcome is blocked, but in the presence of suppression, adding a mediator can increase the coefficient on exposure. Suppression occurs when there are two compensatory pathways from the exposure to the outcome, and holding one of them constant magnifies the other. For example, the effect of doing cardio exercise on weight. Consider the mediator "number of calories consumed". Doing cardio increases the number of calories consumed, which increases your weight, but it also increases the number of calories burned, which decreases your weight. If these effects were equal, there would be no total effect of cardio on weight. However, holding constant one of the factors, it is clear the size of the effect would increase. For example, holding constant number of calories consumed, the effect of cardio on weight may be quite large (in the negative direction); indeed, this is the primary motivation behind controlling your diet when exercising to lose weight. Suppression is described in detail by Kim (2019).
The coefficient on exposure represents the partial correlation between the exposure and outcome given the covariates in the model. So if including a confounder decreases the correlation between the exposure and the outcome, the coefficient on the exposure will also decrease. As far as I know, the coefficient of the exposure tells us the effect of the exposure on the outcome when other variables are held constant. It does not reflect degree of association between the exposure and the outcome This is incorrect. The coefficient on the exposure does represent the association between the exposure and the outcome; that's all it represents without strong assumptions about the nature of confounding. It may be a partial association (i.e., after adjusting for other covariates in the model).
For an excellent introduction to the interpretation of linear models in the context of causal inference (i.e., with respect to mediation and confounding), I recommend Pearl (2013).
Best Answer
The coefficient on X in Y ~ X is the total effect of X on Y. The coefficient on X in Y ~ M + X is the direct effect of X on Y. The total effect is the sum of the direct and indirect effects, where the indirect effect is the effect of X on Y through M.
In your situation, there is no total effect of X on Y, but there is a direct effect on X on Y. This can occur when there are two opposing causal pathways from X to Y. For example, it has been observed that wearing a helmet does little to prevent injury to cyclists (i.e., the total effect of wearing a helmet on injury is zero). This could be explained by the fact that helmets encourage riskier behavior by cyclists (i.e., because they feel safer), thereby increasing the risk of injury, while the helmets themselves provide safety to the cyclists, decreasing the risk of injury. The indirect effect (the effect through risky behavior) and the direct effect (the effect due to the safety provided by the helmets) are in opposite directions, yielding a total effect of zero, even though there is a nonzero direct effect.
Of course, what you observed has occurred in a sample, and so failing to find a significant effect doesn't mean there is no effect. It just means you might not have the precision to detect it. Direct effects can often be estimated with much more precision that total effects, so it could just be that your estimate of the total effect isn't precise even though a total effect is present.