My understanding of the rule is this: "that" identifies, "which" describes. So -

"The car that is red is for sale."
"The car, which is red, is for sale."

In your 'labour pains' example I'm inclined to side with you because I can't imagine that there are any other labour pains being discussed. Therefore, they don't need to be identified, just described. So 'which brought forth the child' seems (and sounds) right to me. The labour pains might have stopped, after all, and not yet done what they're supposed to do - so, again, they require description not identification.

The Economist style guide says "Americans tend to be fussy about making a distinction between which and that. Good writers of British English are less fastidious." - So you're in good company.

My copy of The Times Guide to English Style and Usage (1992) bans 'amongst' and 'whilst' - but Fowler (1965) is less fussed.