Dear Francesca,
Thanks for your post.
The origin of the peak you witness at around 65 keV in the DETECT spectrum can be traced back to additional gamma lines between roughly 32 keV and 36 keV from the relaxation of 137mBa following the beta decay of 137Cs. You see them summed up in your DETECT spectrum, since it does an event-by-event energy deposition.
You do not see the peak corresponding to the sum of the two lines anymore when you put the source 5 cm in front of the geometry because the solid angle subtended by your detector drops drastically and the chance to capture both of these gammas drops as well.
Hope this helps,
Cesc