This is all helpful. Do either of you (or anyone else reviewing this later) think I'd have a shot at admission into a Master's program at one of the schools I mentioned? There are a few points that I think are lost in translation at the moment which I want to address, because I believe my understanding of Statistics is better than you may perceive it to be.
For slightly more context, I am working in an engineering science research lab now, and I am very aware of what goes into conducting top-tier research in science. Statistics and mathematics is more guess work for me because it is the area I'm weakest in. But statistics is the field I want to work in.
To both of your points, I think I do understand what a PhD in Statistics signifies, and why there is a distinction between Statistics and Applied Statistics among departments. In my mind a PhD statistician from a top university is someone who has expertise in core statistical concepts, such as stochastic processes, sampling theory, estimation theory, and decision theory from both a frequentist and a Bayesian perspective; has graduate level exposure to linear algebra, analysis, and mathematical probability; and has advanced exposure to algorithms and data structures for scientific computing. (I think the quality of training in algorithms and scientific computing is the hardest to estimate among these categories, but it is clear that it is advanced.) Then of course there is the actual subfield that a candidate studied in their dissertation or presently pursues. Imaginably, a PhD will combine their statistical expertise in their area with practical knowledge of how applied research is conducted and what policies and regulations govern the same.
My goals for obtaining a PhD in Statistics are two-fold: one is to learn the specific area I'm interested in for a long-term career, and the second is to develop expertise in the topics mentioned before. The area I want to study is relatively old, but has rapidly expanded due to advances in computing technology. I suspect that it will become much more significant as we continue developing new sensors for measuring and logging data that the field addresses, especially as the market for those technologies grows in private industry. In that time, I think there will be an abundant need for people with expertise in that area in academic, government, and industrial settings, and I would be interested in making a career collaborating with people and organizations in each of those respective settings. My greatest preference would probably be to continue to a Postdoc or Assistant Professor position, but I would not turn away from the right industry or government position.
The place I'm at now is a dead end for the career I want to have. It is a top-tier research environment in an area that I have no interest in pursuing, and there are few perceivable opportunities to transition into a better place from here. I don't want to become a business analyst or a data scientist or a machine learning guru—I want to become a statistician. Realizing how bad the circumstances I'm in now are, I am frustrated that I allowed myself to get here. By next year I want to correct course and start in a program that has higher odds of leading into the area I want to research in.