Before proceeding, I would like to mention two complementary materials to the paper that you are mentioned, where you can see significative a diversity of ideas concerning causality.
(1) Gorard, Jonathan. Algorithmic Causal Sets and the Wolfram Model. arXiv preprint arXiv:2011.12174 (2020). ,
(2) Max Piskunov's Bulletin Confluence and Causal Invariance.
The first remark is that in reference (1), which is basically a paper about causality, the phrases Minkowski lattice and planar embedding are not used anymore. We see some evolution in the development of the theory. Personally, I do not like Minkowski lattice.
A yes/no answer to your question is "no, causal relation does not dependent on planar embedding". Indeed, we could imagine a spacetime that is the surface of a torus, where one dimension is a one-dimensional space (circumference) and the other dimension is a cyclic time (circumference). Locally, this spacetime looks like ordinary spacetime and the local causal graph is acyclic. Nevertheless, the surface of a torus cannot be embedded in a plane and the global causal graph contains cycles.
Even considering this question only locally, the combinatorial structure of the (local) causal graph should remain invariant under the composition of the embedding with a Lorentz transformation: this is the essence of special relativity.
The core of the Wolfram Model is what was written by S. Wolfram in A New Kind of Science and in the Technical Introduction to the project and complemented in his writings. The ways in which S. Wolfram's ideas are mathematically formalized and explored by other researchers may change with time. Therefore, if you are interested in learning about causality in the Wolfram Model, my advice is to read S. Wolfram first, e.g.,
The Phenomenon of Causal Invariance
The Role of Causal Graphs
Event Horizons, Singularities and Other Exotic Spacetime Phenomena
Faster than Light in Our Model of Physics: Some Preliminary Thoughts
and then, after you are familiar with S. Wolfram's original ideas, you could read other authors, like references (1) and (2). This is just a personal opinion, other people may have other opinions.
