Question: at what point does "specializing" in one or the other make you not a "full stack" developer anymore? I can technically do about everything on the web, I'm just really bad at front-end, and really good at back-end, which is where I do most of my work. Am I "full stack"? How good to you have to be in...well, everything, to call yourself "full stack"?
But I digress.
Even in the common context, to be Full Stack, I'd argue that you should be better than the sum of the parts. The lift you get from being able to decide, intelligently and at the speed of thought (without talking to a counterpart on the other end), which pieces of code - optimization, safeguards, business logic, etc. - belong in the front and back end, and how they interact at peak effectiveness, is more valuable than just the ability to build a good API or build a good web page.
Further, a full stack engineer, in my experience, is more likely to reach for tools like websockets, memcached, etc. earlier in a project than typical REST APIs (for example) where the line between back- and front- end require a bit more coordinated architecting, to achieve a richer end result.
But I doubt this is widely accepted. Anyway, my point is I wouldn't focus on either just because; I'd focus on what makes you better by knowing some of both. What can you learn in the back-end to support problems you're familiar with in the front-end, and what can you do in the front-end to make better use of your knowledge in the back-end? That's where full-stack value really lies.