Most ATFs contain some combination of additives that improve lubricating qualities,
[1][2][3] such as anti-wear additives, rust and corrosion inhibitors, detergents, dispersants and surfactants (which protect and clean metal surfaces); kinematic viscosity and viscosity index improvers and modifiers,
seal swell additives and agents (which extend the rotational speed range and temperature range of the additives' application); anti-foam additives and anti-oxidation compounds to inhibit oxidation and "boil-off"
[4] (which extends the life of the additives' application); cold-flow improvers, high-temperature thickeners, gasket conditioners, pour point depressant and petroleum dye.
All ATFs contain friction modifiers, except for those ATFs specified for some Ford transmissions and the John Deere J-21A specification;[5] the Ford ESP (or ESW) - M2C-33 F specification Type F ATF (Ford-O-Matic)
So in brief, each application (TRANS - P/S) uses it's own lubricant and the usual rule is to use what the manufacturer recommends. You only use another style lubricant if the system has been modified from OEM and/or has been modified and calls for a different lubricant.