I think it would behoove you to read the rest of the post. The double quotes are not the operative part of example there; they're only there to demonstrate that the code execution doesn't come from splatting or word splitting.
The actual code execution in Case #1 comes from the fact that bash (and other ksh descendants) run arithmetic evaluation on some strings in arithmetic contexts, regardless of their double or single quoting. That evaluation, in turn, can run arbitrary shell commands.
