What is the best way to phrase and punctuate this type of "this = that" sentence?
No noise was detected, indicating that the room was empty.
No noise was detected, which indicated that the room was empty.
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Both of your sentences are phrased and punctuated correctly.
Here's a couple of alternatives, removing "that":
No noise was detected, indicating the room was empty.
No noise was detected, which indicated the room was empty.
I prefer the first of these alternatives, since it's a bit simpler (as BlackVegetable mentioned).