I think when the term was first employed it was referee's assistant but as everyone soon started transposing this to assistant referee the latter became adopted by usage.
As for Scott Oldham, I shall have the opportunity to see him for the first time at Charlton on Saturday.
I think (so could be wrong) that Assistant Referee has always been the correct "title". When the job title changed from Linesman, some of our well paid (but no so well informed) commentators and pundits transposed the words and the term "Referee's Assistant" gained some currency.
I do believe that there is a significant difference in the respective meanings:
Assistant Referee / manager / prime minister / surgeon = a person who is junior to the main job holder, but one capable of stepping in if needed.
Referee's / manager's / prime minister's / surgeon's
Assistant = a person in a support / administrative role who enable the main job holder to focus on the job in hand, but not capable of stepping in if needed.
The two people carrying flags are certainly Assistant Referees.