Your explanation is correct.
Negative and positive current simply refers to the direction of the current, not the magnitude. It's designation as positive or negative is arbitrary. Thus, in the simulation, they chose to have the current out of the battery and through the transistor be negative. Sometimes simulators use current into a node as positive, thus current out of the positive battery node, which is normal, would be negative. If you were charging the battery, the current would be positive. If you look at the graph, the maximum (negative) current occurs at the maximum input voltage as you would expect. Think of the negative current as sort of a mirror image of the positive current.