Yes, it makes a difference. Geometric thickness is what the OP is asking about, while thickness is a related but different parameter (since the embeddings do not have to be the same).

I think the minimum number of colours is sometimes called geometric thickness.

It is not always possible with two colours since an $n$-vertex planar graph has at most $3n-6$ edges, and the red subgraph and blue subgraph are both planar graphs. So, any $n$-vertex graph with at least $6n-11$ edges is a counterexample.

I think it is correct as is. For example every $n$-point $\ell_2$-metric can be embedded in $\ell_2$ with dimension $O(n)$.