I feel a person can be a "a civilized person" without being a "cultured person".
There is a big difference between those definitions.
A civilized person is polite and courteous; he knows how to say "please" and "thank you."
I have met a lot of "cultured persons" in my lifetime. A cultured person can be someone who is highly educated, has good manners, is an avid reader, who watches classic films, listens to classic music and who has a refined appreciation for
art.