I have been trying to define $A\overset{\vert}{\smile}_{C}B$ in a strongly minimal theory (let's say countable to avoid complications though I'm not sure if this matters). My attempt is based on the (perhaps incorrect assumption), that in this setting there is a fairly simple way to do this using the properties of $acl$ and not focusing on the huge machinery associated to forking (I'm just starting out in stability and I don't know much of the specifics, so as I said, I could possibly be completely wrong).
My first attempt was to define $a\overset{\vert}{\smile}_{C}b$ iff $a\in{acl(Ab}$ \ $\{a\})$. However I had no luck proving symmetry with this definition (and this definition is probably wrong as $acl(A)\overset{\vert}{\smile}_{A}B$ does not look like it would hold (even though I don't have a counter example)).
I then tried $a\overset{\vert}{\smile}_{C}b$ iff $a\in{acl(Ab)}$ \ $acl(A)$. Now symmetry is immediate due to exchange. However transitivity is not easy to prove (in fact it is probably false).
So my question is, (if possible) how do you define $A\overset{\vert}{\smile}_{C}B$ in a strongly minimal theory with just the "basics" (say Lemma 6.1.3, Lemma 6.1.4 of Marker's text)?
Also as a side question, how do you get a nicer symbol than $A\overset{\vert}{\smile}_{C}B$? I found https://tex.stackexchange.com/questions/42093/what-is-the-latex-symbol-for-forking-independent-model-theory online but I don't know how to adjust the code to run on stackexchange.
I'm going to use the same approximation to the forking symbol you did, since I don't know how to make one on stackexchage either.
First, let's agree that $A\overset{\vert}{\smile}_{C}B$ if and only if $a\overset{\vert}{\smile}_{C}b$ for all finite tuples $a$ from $A$ and $b$ from $B$.
Given a tuple $a$ and a set $D$, we can define $\text{dim}(a/D)$, the cardinality of a basis for $\text{acl}(a) = \text{acl}(aD)$ in the pregeometry determined by the strongly minimal set when we allow parameters for $D$. Note that enlarging $D$ can only make $\text{dim}(a/D)$ go down.
Now we can set $a\overset{\vert}{\smile}_{C}b$ if and only if $\text{dim}(a/Cb) = \text{dim}(a/C)$.
In the special case that $a$ is a single element, $\text{dim}(a/D) = 0$ or $1$, so we get $a\overset{\vert}{\smile}_{C}b$ if and only if $a\in \text{acl}(Cb)\setminus\text{acl}(C)$.