The logic behind the first case is not applicable in the second case. This is because in the first case, the mirror works on the principle of reflection of light, whereas in the second case, the lens works on the principle of bending of light.
The correct logic is that converging systems have a real focus while diverging systems have a virtual focus.
In the first case, a concave mirror acts as a converging instrument. Therefore, it has a real focus. On the other hand, a convex mirror acts as a diverging instrument. Therefore, it has a virtual focus.
In the second case, a convex lens acts as a converging instrument. Therefore, it has a real focus. On the other hand, concave lens acts as a diverging instrument. Therefore, it has a virtual focus.