The 20th century was the century of the child?

We are in the 21st Century.

It probably was as lots of new technology had been invented then.

it went from 1900 to 1999 it was a span of 100 years which is also known as a century.

Different people would interpret this differently, and there is no generally accepted rule (as the fact that you have asked the question indicates). Of the two suggestions you offer, I would prefer...

I don't think there were any movies in the 19th century or the 1800's