Switches have contacts. Contacts are essential to makemcurrent flow. If engine is not started then whole power for bringing up the windows come solely from car battery. If contacts are not good and they are dirty, that produceses high resistance on the switch which will need more current to create same power. Your car battery mightmnot be up to that job. When engine is running thenmalternator produce power needed for switch to work even if it is dirty.
No, there is no difference between Si or ti. If you have problem with one window and with same switch on the other car you do not, then you might have problems with either motor on that window (not likely), or plastic rollers which moves on window mesh. Rollers are plastic and might be on its way out, thus producingmresistance they should not.