While I see what you are saying, the OP to your post has it correct. You can say a republic is a bastardized form of democracy, but the reverse is not true. You can look at one of the few true democracies that has ever existed as an example - ancient Greece. Every citizen[1] could vote and put forward changes to laws or even create new laws (obviously with a majority vote).
[1] A citizen was considered to be any person born of Greece, white and male.
[1] A citizen was considered to be any person born of Greece, white and male.