But this is not (I think) how most people would speak, and I wouldn't say this. I would say the subconvexity problem is the problem of bridging the gap between 25% (convexity) and 0% (Lindelof). Subconvexity bounds for various L-functions (zeta, Dirichlet L-functions, and higher rank L-functions) are a very active area of research, and so it seems strange to me to say "the subconvexity problem is solved."
"My understanding is that Nelson has indeed done this for every L-function." One minor note. Nelson's work applies if the L-function is automorphic. By the Langlands philosophy, this should be true for any reasonable L-function, but this is far from known, even in, e.g., the simple case of the L-function corresponding to the tensor product of two automorphic L-functions.
Edit: I am wrong about the statement "this is not (I think) how most people would speak." It looks like beating the convexity bound at all is often described as "solving the subconvexity problem," e.g. in the introduction here https://link.springer.com/article/10.1007/s002220200223. This description is a bit strange to me, but if it is standard then it is unfair for me to call it an "inaccuracy," persay; thanks for pointing this out.
The first, pointed out already by gavagai691, is that it is not known that "every L-function" is "standard" (this is a major open problem in the Langlands program).
The second is that the general "subconvexity problem" asks for a nontrivial estimate not only for each fixed L-function as its argument varies, but more generally as the L-function itself varies. The general case of the problem remains open.
My understanding is that Nelson has indeed done this for every L-function, although the exact x varies with the L-function. How is that not solving the subconvexity problem?