Questions on estimating causal influence from Directed Acyclic Graphs

Question

I am reading McElreath's Statistical Rethinking book and I'm wondering if anyone can help clarify my doubts on confronting confounding with DAGs. I'll specify 2 examples:

1. Taken from link. The DAG below is an example on p.186 of the book. $U$ is an unobserved variable. To find the causal influence of $X$ on $Y$, we find all paths from $X$ to $Y$, classify them as open/closed, and do conditioning if necessary. The path with the node $B$ has a collider so we want to exclude it. The paths that we care about are then given by $X \leftarrow U \leftarrow A \rightarrow C \rightarrow Y$ and $X \rightarrow Y$. Am I therefore correct in saying that to estimate the total causal influence of $X$ on $Y$, I regress $Y$ on the variables $X,A,C$?

2. Taken from link. The DAG below is from Problem 6M3 on p.189. Here, the two non-direct paths from $X$ to $Y$ contain $Z$ which is a collider. In this case, how do we then estimate the total causal influence of $X$ on $Y$? Including $Z$ as predictor in the regression would confuse the model since it's a collider, i.e. all coefficients of the model except that for $Z$ would be conditioned on $Z$ (and some other variables). Do we just regress $Y$ on $X$?

Thanks for reading and I hope to get some clarifications!

Answer 1

For example 1, one of $(A, C)$ would suffice to close the open path, but controlling for both is also fine. In example 2 there are no open paths, so there is nothing to control for (and indeed controlling for $Z$ would be wrong), just as you say.