Maybe I use "not" particle in english wrong sometimes. I think it is confusing for both sides. Just couple of examples to demonstrate russian style:
"Я хочу это" = "I want it"
"Не я хочу это" = "(It is not me who) wants it"
"Я не хочу это" = "I do not want it"
"Я хочу не это" = "I want (something, but) not it". It this last sentence correct? As I see "I want not it" is wrong. To be more clear: "I want something, but I do not want this particular thing".

> So "не так" in this statement is as if the speaker was turning and pointing at the baby and saying 'not like this'.
Yes.

> I'd have wanted to say "вы бы не так заплакали, но еще!"
"вы бы не так заплакали, но еще сильнее!" is more correct. "Еще" is hard particle.

> although I think we English tend to use tone to emphasise things more than Russians?
I do not think so. Because of free order of words in sentence we use tone often. You are correct - "not (word)" is stressed and "еще" too.

> How would one say 'You would not have cried so much'?
I think "Ты бы так сильно/много не плакал".