FWIW, when I applied for the Discover IT student I had to verify my enrollment. They gave a list of options, including photo ID, a screenshot of your schedule, a transcript, a note from the registrar, or a variety of other options. If they say that they require a 4 year degree program and you're going for you AD, I wouldn't risk it. The good news is that with the history you have, you should qualify for the regular version!
When I applied for the Citi Forward student, on the other hand, I didn't have to provide any proof. I just selected the box and got instantly approved. I would double check the terms before you and your wife apply, but assuming your programs count, you should be golden with that!
My wife didn't feel comfortable "lying" on a credit card application. Even though it's not a complete lie, it wouldn't be the truth either. We will just apply for the normal version and see what happens!
BoA 123 $2.5k (11/12) // Discover It $3.9k (7/13) // Citi Forward Student $3.8k (7/13) // Chase Freedom 5k (12/13) // Amex Blue Cash 6k (1/15) Starting Score: No Credit History (11/12) Current Score: MyFico 761 Goal Score: 800+