It's the nipple stimulation from the pump that put her in labor. you can do nipple stimulation without a pump :) your milk does not have to be in to do nipple stimulation. You will start to produce colostrum AFTER baby is born which is their first milk. that will later turn into your full milk. :)
I think I'd personally rather labor just start on its own that milk myself. ha ha. In general, nipple stimulation is what causes the uterus contractions and might bring on labor. You can do that without too much effort but agree that the nipples are awful sensitive during pregnancy. I'd try walking or having sex with your partner which can also bring on labor. good luck
You can try to milk yourself manually. My midwife suggested nipple stimulation. Obviously your milk won't be in until bub arrives but you should get some colostrum.