Total/Gaussian curvature is intrinsic, yet mean curvature is extrinsic, why? What characteristics define the total/mean curvature to be intrinsic/extrinsic accordingly? What is different geometrically about these curvatures that cause them to be defined as this?

The idea seems to be that Gaussian curvature is _intrinsic_ because it is invariant over isometric embedding. That is, if you apply an isometric embedding to a surface, the mean curvature of the result may be different (making mean curvature an _extrinsic_ property), but the Gaussian curvature (and any other intrinsic measurement) would have to remain the same.