The only specific trains that you are required to take are the trains of the Bahn (ICE/IC/EC) listed on your ticket. Where regional trains (S/RB/RE/IRE) are shown, you can take any regional train on the same route from midnight (0000) until 10:00 AM the following day.
You'll probably see on your printout the statement:
"Gültigkeit: your date
DB: Gilt nur in gebuchten Zügen an angegebenen Reisetagen. Im NV(S/RB/RE/IRE) auf eingetr. Strecke vor/nach den geb. Zügen am 1. Tag u. Folgetag bis 10 Uhr."
Validity: your date
DB: valid only in the booked trains on the specified travel days. In regional trains(S/RB/RE/IRE) on the designated [my translation] segments before/after the booked trains on the first day and following day until 10 o'clock.*
It's called the vor- und nachlauf clause of the Conditions of Carriage.
The printed ticket will not fit into the cancelling machines. Tickets that have to be cancelled will have the words "Hier entwerten", cancel here, printed on one end.