Yes, the clincher is the acronym on the medal (S.S.J.F.A.).
The other cup finals mentioned did not come under the governance of South Side Juvenile Football Association.
On looking into 'Gymnasium Park', the image below is lifted from a 1918 map of Glasgow. The Govan Police Force owned the police recreation grounds shown on the map.
In a previous comment on this thread there is a press mention of another game being played at Moore Park which you have to assume is somewhere within the wooded grounds of Moorepark House, as shown on the map just south of the police recreation grounds. This is clearly named Moore Park in the press cutting so I have to assume that Gymnasium Park is part of the police recreation grounds.
Apparently Benburb FC were based at the police recreation grounds between 1911-1925 https://en.wikipedia.org/wiki/Benburb_F.C.