This new wonderful note, F-singularities: a commutative algebra approach, written by Linquan Ma and Thomas Polstra, two card-carrying commutative algebraists, is perhaps what you need. From the Introduction: "Our approach is completely algebraic, and most of the results are stated and proved in the greatest general form."
$F$-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).