The definition you quoted explains why quantization discards information: multiple distinct input values map to the same output value. Once that happens, you can no longer determine the original input from the output alone. The distinctions between those inputs are the information that has been lost.

I’m not saying that quantization doesn’t lose information. Instead, I’m saying that every step to lose information is not quantization. Functions on the preimage (clipping, masking) are not quantization.