That's the way the brake pad warning light works. The pad sensor wire completes a ground circuit through contact with the brake rotor (usually the ridge on the rotor that hasn't worn due to lack of contact with the brake pad friction material). The harder you push the pedal the better the contact. When the brake is released the light may or may not flicker depending on how much play you have in your wheel bearings and whether you are driving around corners. Although you can go quite a while with the light lit (esp. if the rotors are worn since the light comes on very prematurely) if you drive too long the warning light will cease to opearate since the sensor will be worn all the way through. Also if the pads get too thin the friction material can separate from the pad backing plate causing all sorts of problems.
When you replace your washer pump be sure to replace the rubber grommet that the pump inserts into---these always seem to leak when the pump is removed and reinserted or replaced. Mark