Exactly.

It should have read, "my family are all Democrats or Greens."

Democratic literally means something relating to democracy. A democrat is someone who practices social equality; or when capitalized [ie, Democrat], a member of the Democratic Party of the United States. Also, a Green is a member of the Green Party.

I'm not going to bother going into the Democratic-Republican Party, and the formation of modern day Democrats and Republicans... no one wants my history lectures.

I mean technically speaking, the thread, which is called "democratic or republican" is actually asking about governmental systems, not political parties. But we've gotta live leeway sometime.