Thanks everyone for sharing your knowledge and experience! I think, for now, I won't be able to resist pulling/digging out the weeds. I do try to get to them before they seed, and they never go in the compost. We do not have that withering Texas sun, and the New England summer humidity keeps them alive forever.
I will, however be keeping a close eye on those
gardeners who follow the Master Gardener's direction and cut their weeds. I'll come back and let you know if I learn anything.