Thanks to everyone for the excellent help I've received using this forum. I now have two mystery weeds I'd like identified - more for the sake of being able to curse them by name than any other reason. Both uproot fairly easily, at least on the day of a watering.
The first I've spent several hours pulling out, it's slightly darker than my St. Augustine, and noticeably more rigid, uncomfortable to walk on barefoot. It stays low to the ground and grows out laterally. As below:
The second one is smaller, and gets no more than about five inches in diameter: