Hello Saint J.
I suggest that it would certainly have been 'RAF Kallang' from some time in 1941 until the surrender to the Japanese in February 1942, and from the end of WWII until sometime in (say) 1946 when RAF Changi became useable/operational. During those two periods, it seems Kallang was used operationally and controlled exclusively by the RAF, making it de facto "RAF ..." I would have little doubt that the site would have had a perimeter fence and appropriate RAF signboards etc. I have also seen it reported that RAF Kallang did not revert to civilian control until 1949, though obviously accepting 'civilian' aircraft from late 1945. The Australians seem to prefer the title 'Kalang Aerodrome' or 'Kalang Airstrip.'
This from Wiki - https://en.wikipedia.org/wiki/Kallang_Airport
The Kallang Airport also known as the Kallang Aerodrome, Kallang Airfield and RAF Kallang,
and ............
When the Japanese launched their invasion of Malaya and Singapore on 8 December 1941, Kallang was the principal fighter airfield. By January 1942, it was the only operational fighter airfield in Singapore, as the other airfields (Tengah, Seletar and Sembawang) were within range of Japanese artillery at Johore Bahru.
Brewster Buffalo fighters of 243 Squadron RAF, 488 Squadron RNZAF and a detachment of 2-VLG-V of the Royal Netherlands East Indies Air Force operated from the airfield, defending Singapore from repeated Japanese air raids. They were joined later by Hawker Hurricanes of 232 Squadron RAF, but attrition took a steady toll on men and machines, and by the last days of January 1942, the airfield had been badly damaged by the bombing and only a small number of aircraft were serviceable. The last of the fighters left in early February, escaping to carry on the fight just before Singapore was surrendered to the advancing Japanese.
I am led to believe the runway at Kallang was built by the Japanese, and that the airfield was under the control of the Japanese Army Air Force. The airfields in the north of the island - Tengah, Seletar and Sembawang, were apparently controlled by the Japanese Navy. As Changi was also north of the 'Army/Navy demarcation line' perhaps it too was Navy. I got the impression Changi was barely operational by war's end.
HTH
......... LFH