Boxplot whiskers for a journal publication using APA style guide
Major statistical software such as Stata, SPSS, SAS, and R all use John Tukey's definition (hyperlinks are to the related technical documents), with various degrees of customizability. Some of them also cite William Cleveland's scheme, which is the same as Tukey's.


  one standard deviation above and below the mean of the data


This does not make sense as box plot is largely a device to display the central tendency of data using the more robust percentile statistics and I have a hard time pairing it with standard deviations (and its related statistics standard error and 95% CI), which based on the actual value of the data rather than their rank. If showing mean and SD (or SE) is the motive, there are error bar plots for this purpose.


  the minimum and maximum of all of the data


While min and max also indicate spread, they are also sensitive to extreme values, and hence a less robust indicator of dispersion. So also not making much sense.


  the 9th percentile and the 91st percentile &amp; the 2nd percentile and
  the 98th percentile


I have less opposition against these schemes, the only reason I'd opt for not using them is just we don't need any more schemes.

So, my bet is on Tukey's. And if you're uncertain if readers may be confused, by all means in the footnote state that the scheme is based on Tukey's box plot.