"A confidence interval gives a range for E[y∣x], as you say. A prediction interval gives a range for y itself.".

In the vast majority of the cases, what we want it the range for y (prediction interval), that is, given x = 3, what is the expected distribution of y?. For example, say we train a model to estimate how the 100-m dash time varies with age. The uncertainty we want is, "at age 48, 90% of Master Athletes run the 100-m dash between 10.2 and 12.4 seconds" (here there would be another difference to point out between Frequentist and Bayesian intervals, but let's make things simple).

We are generally not interested in, given x = 3, what is the uncertainty of the expected value of y (that is, the confidence interval)? In this case, the uncertainty we get (we might want it, but often we do not), is, "at age 48, we are 90% confident that the expected time to complete the 100-m dash for Master Athletes is between 11.2 and 11.6 seconds".