The carbon-13 NMR for 1,1,1-trichloroethane
I fully agree with your assessment and assume that this is a mistake in the book. I wonder what exactly the authors suggested as spectroscopic verification. I personally would suggest using 1H-NMR as a verification method. In 1,1,1-trichloroethane, only one hydrogen peak should be observed as the three protons of a methyl group are always homotopic. In 1,1,2-trichloroethane, two signals in a $1:2$ ratio should be observed as a triplet and dublet, respectively. Depending on what the book suggests as verification, it should be easy to confirm the nomenclature error.