Characterisation of Turing completeness?
It seems very unlikely that there is any useful characterization other than the definition. Any such characterization would have to deal with all kinds of programming languages (imperative, functional, declarative and everything else) along with, assuming the claims from the Wikipedia article you link are correct, Minesweeper, Conway's game of Life and Magic the Gathering.

If you did have a characterization, it would probably be no easier to apply that characterization than to just apply the definition.