The classic definition that medievalists would use is that the Scholastics (literally "Schoolmen") were those who studied and taught in the theological faculties of the universities like Paris, Oxford, etc. and who wrote in the expository style called quaestio. Not all Scholastics were friars - members of the Domincan and Franciscan orders - but almost all were and this is also traditionally considered a mark of Scholasticism.
Many medievalists see Peter Lombard as the first true Scholastic - he taught at Paris and he compiled a textbook called The Sentences in the 1140s which, along with the Vulgate Bible, Plato and Aristotle, was the main source work that all the great Scholastics studied and wrote commentaries on.
The Summa Theologiae, considered the greatest work of Scholasticism, was an attempt to create a new textbook for beginning theologians that would surpass The Sentences.
Since Anselm and Eriugena were not attached to any university faculty, were not friars and wrote in dialogue form and not quaestio form they are usually not considered true Scholastics, but important forerunners of the intellectual movement centered in the high medieval university. Anselm especially, because he was focused on logical analysis and because he corresponded with university lecturers.