@the person who downvoted this: it will be much more constructive to write a comment to explain your issue with this answer.

It certainly is. Was there a downvote? Wow!

At least for the two dimensional version of this problem, the minimum is not the same. Take a 3x3 grid in the plane. The minimum number of lines for S is 4, while T can be covered with 3 lines, x + y = 1, x + y = 2 and x + y = 4. I guess it can be generalised to higher dimensions as well, but for that I will need a pen and paper.

@DylanThurston: I believe these are some crucial papers in this history of CN. Alon-Friedland-Kalai, Regular subgraphs of almost regular graphs (84); A-Tarsi, A nowhere zero point in linear mappings (88); A-Tarsi, Colorings and orientations of graphs (92). A-Furedi, Covering the cube by affine hyperplanes (93); A-Nathanson-Rusza, The polynomial method and restricted sums of congruence classes (96); and finally the paper titled Combinatorial Nullstellensatz by Alon from 1999. In the Alon-Furedi paper it is even mentioned that Hilbert's nullstellensatz can be used to solve the main problem.

@Seva: There is a multiplicity version of Combinatorial Nullstellensatz as well. See Theorem 3.1 in S. Ball and O. Serra, Punctured Combinatorial Nullstellensätze, Combinatorica 29 (2009), 511-522.