Command substitution and double quotes: why results are different?

You are right, it is something else in this case.

The solution is still in the same link, but the second point:

• Nested quoting inside $() is far more convenient.

  [...]

  `...` requires backslashes around the internal quotes in order to be portable.

Thus,

echo "`echo \"test\"`"

does not equal this:

echo "$(echo \"test\")"

but this:

echo "$(echo "test")"

---

You need to compare it instead with this:

echo "`echo \\"test\\"`"