Why Altium routes trace to the side of the pad?
It's worth noting that the mouse-snap actually uses the autorouter to some extent. 

Depending on the routing mode you have enabled, it will behave differently. Shift+ R while in the interactive routing mode (e.g. placing traces) changes the routing mode.

Anyways, if you want to change the orthogonality of the traces as you're routing, you just press the space, and it should try the other angle (note that some routing modes are more responsive to hitting space then others in the more recent versions, particularly if you have pin swapping turned on).