Montiaceae Raf. is the correct and accepted scientific name of this family of plants. It was named and described by Constantine Rafinesque in Annales Generales de Sciences Physiques in 1820. It is commonly known as the Miner’s Lettuce Family.
Plants of the World Online lists 14 accepted species in the genus (as of 4-13-20 when I am writing this page). That number could change periodically as updates are made.
The only species in this family I have identified on my farm in Windsor, Missouri, in Pettis County (Henry is across the road) is Claytonia virginica.