Should I do a PhD or Master in Statistics for a career in Data Science?

I am a Mathematics student with a concentration in Applied and Computational Mathematics. I have done a Mathematics research project with a faculty member, which resulted in a co-authored paper being refereed. This project was on image processing, and I was exposed to some large scale data analysis. The knowledge discovery element of the project was very exciting, and that's what made me decide to go to graduate school.

However, I had a hard time deciding what I want to study. I am naturally interested in many things (I was an Art History major before switching to Mathematics), and not inclined to put forth serious intellectual effort in anything that would narrow my career possibilities afterwards. For example, for some time I was very interested in Financial Mathematics, but then realized that this may limit my career options to the finance industry. With my research experience, I could see that Data Science is an interdisciplinary subject with potential application in a vast number of fields. So Data Science is a possible "major" for my graduate study.

As for the programs, Master programs bearing the names of "Data Analytics", "Data Science" and the like don't appeal to me as much as Statistics MS/PhD with Data Science track. The reason is that many of the former ones seem to be more or less "fad" programs (I'm not wishing to offend anyone here, so correct me if I'm wrong) designed to meet increasing industry demands for skilled data analysts, and so may lack the systematic, deep mathematical rigor and generality (i.e., not confined to business applications) I appreciate in the latter programs.

So it seems a PhD/Master in Statistics would suit my criteria for (1) a widely applicable, interdisciplinary subject and (2) more rigorous, mathematical training of the subject. What do you think?

Please note that as of now I have no particular preference for industry or academia after program completion. Which of the two I will end up in will depend largely on how my job search goes. I know that the number of faculty positions is much fewer than the number of PhD students wishing to fill those positions; also industry may pay better, and I am not dead set on producing academic papers in prestigious journals. As long as I get to work on interesting problems, using data science as a tool, and make some contributions, I would be happy.