Drummond's paper discusses the right way to present error bars graphically. The paper mentions that the confidence interval of the mean should be used instead of standard error of the mean when comparing means is the goal. The paper also mentions the confidence interval of the population, but fails to describe the difference between population and mean CIs. I have not been able to find descriptions of these differences or formulas used. Is the CI of the mean the "regular" definition of CI based on Z or t distributions? If so, what is the CI of the population? Presumably the distinction is not the same as SD of the sample vs SD of the population.
The Sample Mean
If we were to repeatedly draw samples from the population, calculate the mean and construct $(1-\alpha) \cdot 100\%$ confidence intervals each time, then $(1-\alpha) \cdot 100\%$ of these intervals contain the population mean.
Here, the confidence interval is a measure of precision of the estimate. The narrower, the more precise our estimate.
Although I wouldn't call this a 'confidence interval' for the exact reason you are confused, the range described here contains $(1-\alpha) \cdot 100\%$ of the values in the population.
Here, the interval is a measure of dispersion around the true mean. It describes where we expect most values to be.
As ocram mentioned, a better way to call this is a prediction interval.