This question might be vague, or possibly might not have any answer. I am looking for an alternative definition of geometric intersection number. Recall that we have the following standard definition:
Let $\Sigma$ be a orientable surface without boundary and $\alpha,\beta:\Bbb S^1\to \Sigma$ be two closed curves. Define $$|\alpha\cap \beta|:=\begin{cases}\#\big\{(z,w)\in \Bbb S^1\times\Bbb S^1:\alpha(z)=\beta(w)\big\} &\text{ if }\alpha\not=\beta,\\ \frac{1}{2}\#\big\{(z,w)\in \Bbb S^1\times\Bbb S^1:z\not= w\text{ and }\alpha(z)=\alpha(w)\big\}&\text{ if }\alpha=\beta.\end{cases}$$ Now, the geometric intersection number is defined as $$i\big([\alpha],[\beta]\big):=\min_{\alpha\sim \alpha',\beta\sim\beta'}|\alpha'\cap \beta'|.$$ Here, $\sim$ denotes free homotopy, and $[\ ]$ denotes the free homotopy class.
Recently I got another definition, might be equivalent to the standard one, using liftings $\widetilde \alpha,\widetilde \beta$ of the curves $\alpha,\beta$ in the cover corresponding to the subgroups $\langle[\alpha]\rangle,\langle[\beta]\rangle$ of $\pi_1(\Sigma)$ and then considering the full pre-image of $\text{image}(\widetilde \alpha)$ in the universal cover, and then projecting this full pre-image in cover corresponding to $\langle[\beta]\rangle$, and then counting the number of intersection points.
Can anyone suggest to me a reference to this new definition? Any other alternative definition is also welcome.