but whenever my mum cries I don't feel particularly sympathetic or caring, I just feel uncomfortable.
maybe this is because I've seen her use crying as a manipulative tool
Yep , same reason I don't always melt when she cries. Also this thing about being good to your mother only applies to someone who acted like a mother , with empathy and caring. When I see my mother cry over little things I think hypercrit, she use to beat us for crying "Boys don't cry" , I've not cried since I was 10 years old, wish I could now, people seem to think I'm bullet proof.
This addage of being a good person if you love your mom; Mafios people are always good to their mothers (Kray Twins), Jimmy Savil (prolific pedofile) was good to his mother, Dr Shipply Europes biggest serial killer was good to his mom. The movie Psyco etc...