I assume that you are referring to this;
http://www.bahn.de/p_en/view/booking...services.shtml
I have no direct experience of claiming so I can't be sure, but my interpretation is as follows.
If your train is scheduled to arrive 20mins+ later then you can take;
- The next available train (any route)
- Take a later train (with less or no delay)
- Use a different train which could be higher quality (IC/EC/ICE). The supplement would have to be paid and then claimed back
I would say that the 20mins refers to the arrival in your final destination not any intermediate or transfer stations.
I would also say that the train has to be advertised with a 20+ min delay. So if it is stated as 15mins late but you expect it to arrive at your station 20mins then you can't take an earlier train. Not until it is officially schedule to arrive 20mins late.
There are also more restrictions such as "...regulation does not apply to tickets for a substantially reduced fare (e.g. Schönes-Wochenende-Ticket, Laender-Tickets)."
If you are not sure, ask some staff at the station.
Also take a look at the claim form and consider "how would I fill this in".
http://www.bahn.de/p_en/view/mdb/bah...2_20131227.pdf
If you think you can fill it in clearly enough, then you should be OK. Note that you have to record the train numbers in your claim.