Interpreting prediction intervals, and prediction intervals for a specific parameter?

Can someone correct my thinking if I'm off course here?

Confidence intervals provide an estimate of precision$$^1$$ for a specific parameter, but they can also be used for a regression equation (i.e., they signify the degree of uncertainty about the "true" location of the fitted line). As a consequence, they can also be applied to specific predictions of expected values (i.e., if we were to repeat this procedure an infinite number of times, 95% of the time the interval would contain the "true" predicted value for all units who have a specified X score).

Prediction intervals, on the other hand, provide estimates of precision about individual scores. In other words, under what range can I expect one unit's score to fall? As I understand it, a 95% prediction interval means that, over repeated samples, we can expect this interval to contain a unit's actual value? (See Can we make probabilistic statements with prediction intervals?)

With that being said, it makes sense why software algorithms report confidence intervals for statistics and only allow prediction intervals for regression equations. But does it make sense to have a prediction interval for a parameter?

In my mind, it seems like it should not. The only situation I can think of where a prediction interval for a parameter is of interest is if one were to compute the prediction interval of the mean of a specific group. However, in that case, the mean itself also constitutes the predicted value of someone in that group, so it's not the parameter that's of interest, per say, but rather the prediction of an individual belonging to that group.

Am I interpreting these correctly? Thanks in advance!

$$^1$$ I'm not sure if anyone would object to me calling a confidence interval a measure of precision. These interval estimates of precision are to be interpreted as the probability of overlap with the true parameter over repeated samples (e.g., over repeated samples, 95% of the time the interval will contain the true parameter). In my mind, that means CIs are a measure of precision, but that's beside the point.