Example of not computationally indistinguishable
The usual way of proving that two distributions are distinguishable is by devising an attack such as the one you mentioned. Finding an attack may not be easy or even doable. In general, you should assume that two distributions are distinguishable unless you can prove otherwise. To prove indistinguishability, you can often use the technique of reduction to map the distributions in question to other ones that are known to be indistinguishable from random.