Ridiculous.

The definition I cited: ""control of an organization or group by the majority of its members."
The definition you cited: "a system of government by the whole population or all the eligible members of a state, typically through elected representatives." You somehow feel that the involvement of elected representatives is a requisite of democracy, but that's not what your definition says.

So no substantive difference in either definition. In both cases the essence of democracy is majority rule.