Actually all of those definitions explicitly or implicitly only talk about values in the range of the function. The definition of range that I'm familiar with requires that there actually exists an input mapping to each value in the range.
What is interesting though is that in both Pre and aPre y is not chosen uniformly from the range but as the hash-value of a uniform input. That can cause interesting biases in the distribution of y. But I'm not sure if it actually makes an interesting difference.