This is going to drive me nuts, Lin. My chickweed is all tossed, too hot for it now. I am really only familiar with Stellaria, not sure about Cerastium since I haven't seen it in years. But I looked for it in my yard yesterday and again just now and there's nothing left.
I cleaned out the bed where it usually grows, so did my yard guy when he weed eated.
I just can't tell from your pictures but does it mat at all? Mine mats, falls over, but Trish said hers grows mostly in clumps, so I don't know for sure.
No help, am I?