Why do the two cycloheptadiene isomers both show four distinct shifts in carbon-13 NMR?
There are indeed 4 carbon environments in both of these molecules, shown below;

A plane of symmetry can be drawn through each, with one environment having only one carbon nucleus. All other sites have 2 carbons.

I don't know that any words will make it any clearer than the picture, but if you aren't sure after looking at this, I can elaborate.