That definition is a bit too esoteric for my taste.
In the most COMMON interpretation, OA was an earlier ANSI term for the cooling of a transformer NOT being required to have fans to facilitate it. And GoldDigger was correct as to the definition. OA = Oil/Air, meaning the cooling method is oil dissipating into air naturally by convection, so the transformer would just have a tank and a radiator, with the oil circulating in the radiator by virtue of the heat rising from the radiator due to convection, pulling oil in the bottom and back into the transformer from the top. In that older ANSI nomenclature, the choices in oil filled transformers were OA, FA and FOA, where OA was as described above, FA stood for Fan Assisted (for the radiator), and FOA stood for Forced Oil Fan, where there is a circulating pump for the oil inside of the tank, then fan assisted cooling for the radiator. These transformers often came with different ratings based on those cooling methods, i.e. 30/40/50MVA OA/FA/FOA, meaning that with no fans on, it was 30MVA, turn on the fans and you can get 40MVA, then on the circulating pump AND the fans and you can get 50MVA out of it.
The newer (ANSI) IEEE Std C57.12.00 - 2010 designations are a little more comprehensive and separate the cooling medium from the method;
Cooling methods
A = air cooling,
O = oil cooling,
K, L = cooling with synthetic fluid,
W = water cooling
Cooling mode
N = natural (convection) cooling,
F = forced (fan assisted) cooling,
D = directed cooling (directed oil flow)
Then there are two sets of designation letters, the first two are the INTERNAL cooling method, the second two are the EXTERNAL cooling method for the oil. So the older OA designation would now be ONAN meaning Oil Natural cooling internal, meaning the transformer is oil filled, but there is no pump to circulate the oil; then Air natural cooling external, meaning there is a simple heat radiator for the oil on the outside, no fans.