The difference between DFT and DFS
I think part of the problem is an awkward and inconsistent naming convention. There are 4 flavors of Fourier Transforms depending on which domain is continuous or discrete (which maps to being aperiodic or perodic in the other domain). So we have

         Name                  Time                  Frequency
Fourier Transform    continous/aperiodic     continous/aperiodic 
Fourier Series       continous/periodic      discrete/aperiodic 
Discrete Time FT     discrete/aperiodic      continous/periodic
DFT or DFS           discrete/periodic       discrete/periodic


A better naming would have been

         Name                  Time                  Frequency
Fourier Transform    continous/aperiodic     continous/aperiodic 
Fourier Series       continous/periodic      discrete/aperiodic 
Discrete FT          discrete/aperiodic      continous/periodic
Discrete FS          discrete/periodic       discrete/periodic


so that discrete refers to "discrete in time and periodic in frequency" and "series" refers "discrete in frequency and periodic in time". In other words "series" means sums and "transform" means integrals. Discrete mean sums and continuous means integrals.