This new note, F-singularities: a commutative algebra approach, 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).

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).