This question (and the current answers) are out of date. As of 1 October 2016 Deutsche Bahn no longer requires an identification card but only an official identity document (or their Bahncard).

Per their help pages (in German, I was unable to find an English version)

Ab dem 01.10.2016: Identifikation per amtlichen Lichtbildausweis

Bei Buchungen nach dem 01.10.2016 entfällt die Angabe einer Identifikationskarte. Bei der Kontrolle im Zug müssen Sie sich dann lediglich durch Vorzeigen eines amtlichen Lichtbildausweises oder Ihrer BahnCard legitimieren.

Folgende Lichtbildausweise werden anerkannt:

• deutscher Personalausweis
• deutscher Reisepass
• Kinderreisepass
• europäischer Personalausweis
• internationaler Reisepass
• elektronischer Aufenthaltstitel
• Bescheinigung über die Meldung Asylsuchender (BüMa)
• BahnCard (ggf. in Verbindung mit einem Lichtbildausweis)

Es gelten nicht: Führerscheine, Schülerausweise, Truppenausweise und Schwerbehindertenausweise.

which summarizes to what I have stated above. The man in seat sixty-one has already reflected these changes:

If you wanted to use a German Railways print-at-home ticket (shown as online ticket on bahn.de) you used to have to show your credit or debit card as I.D. on board the train. I'm glad to say that this changed in October 2016, all you now need is a passport or other recognised ID to prove your name to support a print-at-home online ticket. You may or may not be asked for it by the conductor.

You have no need to worry. Deutsche Bahn accept debit cards just fine. I have used my UK Visa Debit Card to buy tickets online and identify myself many times.

Furthermore, I believe that when you select "Credit Card" as identification, you have to present the card you booked with(they swipe it to check), not just any card that happens to have your name on it(after all, it's not a photo ID so unless it has the same number it could be the card of someone else with the same name). So providing the website allows you to purchase your online ticket with your card, it is not only certainly valid as ID, it's probably the only card that's valid as ID.

So you want to make sure you have a proper identification, and according to the Deutsche Bahn, your bank card is your best bet. I think the question here is that more what makes your card a debit or credit card.

It depends on how you want it to behave

I have also a bank card that may act as a debit or credit card and have used it extensively either as one or the other, always compatible. On money.SE, the questions I found seem to show that a bank card is not a debit or a credit card, but may act, if the owner decides, as a debit or credit card. Whether this is a debit or credit card determines how the card communicates with the payment system. While this is how you interpreted it for the payment, your question is about the identification. I suppose it is the same, i.e. your card may be identified as a debit or a credit card. My first guess is that if you present your card as a credit card, and the controller uses a machine, the machine will search its credit card features.

DB accepts cards communicating through VISA, MasterCard, American Express, JCB and Diners Club systems

The Deutsche Bahn says it accepts credit cards of the types VISA, MasterCard, American Express, JCB and Diners Club. So the Deutsche Bahn has systems being able to interact with these types of cards, and in the end what matters is that your card is one of these to interact with the Deutsche Bahn system.

DB identifies your card with first name, last name, credit card number and expiration date

The Deutsche Bahn website says that the identification you show to the controller is what you have stated at the time of purchase. And if you follow the booking flow on bahn.de, the credit card option of identification actually asks you for 4 things : first name, last name, credit card number and expiration date. I expect that if your card can communicate with the Deutsche Bahn system, what this system will try is matching this information, no more.

This information is written on any bank card (as far as I know). So this means there is no need for a machine to read your card. The controller may just look at the card and read these. In general, I could not find any information apart from this disputed answer that the controller uses a card reader (the dispute is based on the fact that far from all German ID cards could be read by a machine - and no French ID card, though they are an accepted type if identification).

German debit cards use Girocard, yours probably not

On Wikipedia I could find that German debit cards are run on an interbank system called Girocard. This means that the communication protocol for debit cards (how the debit card and payment system communicate) is Girocard, a German system. So non-German cards a priori cannot interact with the German payment systems (they may, if they are Maestro or V-Pay). That's probably why the Deutsche Bahn explicitly writes that For identification with ec-cards/Maestro, your bank must have its registered office in Germany. So this means that most likely, the controller's machine will not be able to interact with the debit feature of your card, but only the credit feature.

Conclusion

So in the end, while I am no technical expert on the identification of a card, I think that your debit/credit card (as long as it is one of VISA, MasterCard, American Express, JCB and Diners Club) should be accepted as a proof of identification.

An alternative would still be to buy your ticket at any ticket vending machine or at a counter in a station. You would not need any piece of identification. But on some routes, you may pay more buying last-minute at the station than online earlier.