It's not just the tension in the string that is changing. For Arco: when the bow is drawn the equilibrium position that the string is vibrating through is no longer a straight line from the nut to the bridge, due to the pressure from the bow. The deformation of the equilibrium position causes the string to stretch, resulting in higher tension (hooke's law, e.g.) and an increase in vibrating string length. I think the mass of the vibrating string (between nut and bridge) actually increases a bit too, because the tension along the entire string from tuning peg to tailpiece would increase, so the entire string would stretch, but the only location where additional length would be present is in the segment from the nut to the bridge. i.e. some of the string slides over the nut and bridge as the whole string stretches. In principle the tailpiece wire could stretch too, but I suspect that on most basses the tailpiece wire is not as stretchy as the string and there are two wires (a loop actually) that would be picking up the extra tension.
So, IMO, there are three factors: increased string length (which will lower the pitch), increased mass (which will lower the pitch), and increased tension (which would raise the pitch) as the bow pressure is applied and the bow is drawn.
Jim