To be honest, the entire approach of trying to prove which force is older is a bit pointless because it's so open to interpretation and laden with additional nationalistic overtones.
Ultimately, an air force can and should date its founding to the moment from which it was no longer subordinate to the army and navy, as Roger pointed out, but in nearly all cases that exact moment was a little fuzzy and usually predated the creation of the actual enabling legislation. It is further complicated by the fact that different defence forces had varying approaches to command staff structures and made some allowance for inter-service transfers.
In the South African case, the UDF had a rather odd structure for some time, where there was no South African Army
per se, but rather a set of corps, units, and administrations reporting to a Chief of General Staff and Adjutant-General, as opposed to the neat service-oriented structure we see today.
That's the problem with the (IMHO a little mean-spirited)
Australian argument(PDF) that creates an arbitrary set of goalposts and uses it to show how the RAAF 'wins' the age contest. Modify those definitions a little and the winner changes.
So in my view it doesn't really matter who's oldest, who's the oldest Commonwealth/Dominion force, etc, but rather what you as a service choose as your own founding date and can reasonably defend from an historical point of view. From that perspective the SAAF was founded on 1 February 1920, after which General Sir Pierre van Ryneveld technically reported to CGS and didn't have to be subordinate to any units of the traditional army, and it will celebrate its centenary on 1 February 2020 when it should do nothing all extra to try to claim it's the second-oldest anything.
All that being said, there's a fair chance the SAAF will either substantially downplay or perhaps even neglect to celebrate its centenary at all, making this conversation a bit moot. After all, the South African Army opted against celebrating its centenary.