This new note, [F-singularities: a commutative algebra approach](https://www.math.purdue.edu/~ma326/F-singularitiesBook.pdf), written by Linquan Ma and Thomas Polstra, two card-carrying commutative algebraists, is perhaps what you need. $F$-split and $F$-pure rings are discussed starting from Section 2. (of course, there are other sources out there since the beginning of tight closure theory but this looks more convenient).