The problem with the notion of "current" is that it is one of the concepts which is (a) less simple than thought on first sight and (b) often not well explained particularly in introductory textbooks. The reason is that for some (historical?) reason textbook writers think the integral form of Maxwell's equations were the most simple starting point although this is not the case.
It is important to first making clear that a current is a scalar (!) quantity giving the amount of charge per unit time flowing through a given surface. The sign depends on both the direction of the corresponding current density, which is a vector field (!) and the arbitrarily chosen direction of the surface normal vectors across which this current density is integrated to define the current.
This indicates that, as usual, the local notion of the phenomenon under consideration ("moving electric charge") is simpler than the global (integral) one. Starting with the charge density (a scalar field), ##\rho(t,\vec{r})##: if you take a small volume ##\mathrm{d}^3 r## around the position ##\vec{r}##, then ##\mathrm{d} Q=\mathrm{d}^3 x \rho(t,\vec{x})## is the charge contained in this volume at time ##t##. Then if this charge is a fluid described by the velocity field ##\vec{v}(t,\vec{x})##, then the current density simply is ##\vec{j}(t,\vec{x})=\rho(t,\vec{x}) \vec{v}(t,\vec{x})##. Here ##\vec{v}(t,\vec{x})## is the velocity at time ##t## of a fluid element being at position ##\vec{x}##.
Now the current for a given surface ##A## is given by
$$I(t)=\int_{\mathbb{A}} \mathrm{d}^2 \vec{f} \cdot \vec{j}(t,\vec{x}).$$
It's positive (negative) if the flow of the charge is more in (opposite to) the direction of the surface-normal vectors ##\mathrm{d}^2 \vec{f}##. That's all. There's no need for a "conventional current" and a "real current". The signs are all be taken of by these mathematical definitions of the current density vector field, which is of course in the opposite direction than the velocity field, if the charges are negative (i.e., if ##\rho<0##). In addition it's important to keep in mind that the sign of the current also depends on the choice of the direction of the surface-normal element vectors along the surface, the current is referred to.