Is this true or false total solar eclipse can be seen only by people in the area of Earth within the Moons umbra?

True. The umbra is the central darker total shadow of the Moon. Outside of this area, called the penumbra, is the lighter partial shadow. In this area, you would see a partial or annular eclipse. The umbra and penumbra are cone-shaped areas of full and partial shadow.

The solar eclipse of January 14, 2010 (last week, as I write this) the Moon was so far away from the Earth that the umbra didn't reach all the way to the Earth. So the Sun was visible all the way around the Sun; and "annular", or ring-shaped eclipse.