In the past I have received DB tickets by MMS, but when booking a ticket now, I couldn't find that option. The confirmation email says I should print the PDF ticket... is it OK if I just download the PDF on my phone and show that?
According to the rules of transport (Beförderungsbedingungen), section 6.3.3 you can show the ticket on a smartphone, tablet or laptop as long as the backlight is turned on and the barcode is shown in full-size:
Ersatzweise kann in Zügen das Online-Ticket auch auf dem Display eines mobilen Endgerätes über ein pdf-Anzeigeprogramm vorgezeigt werden, wenn der Barcode in Originalgröße und die kompletten Fahrkartendaten bei aktivierter Hintergrundbeleuchtung vorgezeigt werden können. Die Bedienung des Endgerätes nimmt der Reisende vor; das Prüfpersonal kann jedoch die Aushändigung des Geräts zu Prüfzwecken in Anwesenheit des Reisenden verlangen.
Although that section has been in the rules for quite a while, not all conductors know that unfortunately. It adds to the confusion that the PDF clearly says "please print" on several places. A reference to that section usually is enough to convince them otherwise.
Since about a month the official smartphone app DB Navigator also allows to open most usual online tickets directly from inside the app if booked with your bahn.de account. You shouldn't get any problems with using that.
The passenger must show his/her printed Online-Ticket on the train, a file, e.g. on a memory stick, will not be accepted.
Obviously, a memory stick has no screen but, like the confirmation you received, the text on the website is quite clear, the ticket must be printed. The German version also contains similar language.
Note that DB also sells smartphone tickets but it seems to be a separate offering, only available through an app. You could still try to register, download the app and see if your booking is visible in this format but I doubt it will if you did not register beforehand.
A little bit wrong in places. The Deutsche Bahn Regs specifically allow the ticket to be shown as a PDF file on a suitable machine - tablet, laptop etc. - from an email attachment. Two problems though. One is that the code has to be displayed in original size and dimensions. Not all machines do that easily, with mobiles hardly possible. Secondly, some of the local trains are not run by the Deutsche Bahn, but are regional companies and the ticket inspectors there are sometimes 'difficult'. Never had a problem using a PDF on my Android 10" tablet, when travelling on standard trains.