Yeh as already stated, it doesnt matter which one you fly, the only difference is which DME you hold/use. The geographic points dont change, just the point from which you measure the distance from does.
Your choice. Same at Wagga, Melbourne and prob plenty more around the place.
With reference to the approach controller stating "you flew the correct approach", vaguely remember the chart index number changing few months back as 16R ILS got its own paired DME. I may be incorrect... He was prob just referring to that.