Yep, that's roughly right! Atomism first emerged among the Mutazilitie mutakallimun and then was later adopted by the Ashari and Maturidi kalam schools.
It was broadly adopted due to the precieved finatude of events given the impossibility of an actual infinite (Cf. Kalam Cosmological Argument).
As you mention, this atomism is broader than how it was understood by the Greeks; it also holds that the interactions, attributes, space, time and motion are all finite (i.e. discontinuous). It was (and is) still widely held; it forms a key tenent in the 2 of 3 orthodox schools of Sunnī theology. It probably reached its peak arround the 12th c, but there's been a renewed interest but this is of a mixed quality.
The difference between Ashari & Greek atomism has lead some to claim other influcnes, particularly Bhuddist and Byzantine versions of atomism, but this remains unconclsuive and I'm personally unconvinced by this idea.
As such, they naturally also went on to adopt occasionalism, the most famous part of Ashari kalam that holds that the world is being continuosuly recreated and that disavows secondary causality. The (precieved) regularity between causes and events is simply due to the habit of God. As a side note, Mutazilitie mutakallimun did proposed a number of alternatives to occasionilsm whilst remaking atomists (E.g. iqtiran, tawlid) but these were quite quickly rejected.
Decent secondary sources to get started in English would be the Oxford and Cambridge Handbooks of Islamic Theology and Josef van Ess's fantastic The Flowering of Muslim Theology for which there's a good quality English translation available. The Kalam Research & Media journals and publications are also quite a good sources for both historical overveiws as well as the work of contemporary mutakallimun on these ideas.
As a side note the English secondary sources are of limited quality; the best secondary sources are in Turkish and the classical primary sources are in Arabic and Persian but there's also high quality primary work in a range of other Islamicate languages.