I think full stack is just one of those terms which describe someone that works in both front and back end. I consider myself a full stack developer because my job requires me to work in both front and back end. But my strength is in backend code. I think this is true of many developers, they are full stack but their strength is in one of the other.
Also, back end or front end, things change, stacks improve, etc. I do not believe the end exists, I believe you can become very knowledable but every day is a lesson, even for developers who have been coding for far longer.
Some of the developers I work with are incredible, their knowlege in both backend and frontend stuff is amazing, but by their own admission, they do not know everything, which is why forums such as these exist, so we can reach into the community and keep on learning.
More than anything, I believe its down to personal choice. I think a person should do what they feel most comfortable with and makes sense for them and their personal circumstances.