Immigration status

The status is refering to immigration status, which has a close link to the visa. I guess the question is if the VO at US consulate will consider the study permit or work permit as your immigration status or not.

Crossing the order is a different issue. However, the immigration officer at the border has the rights to determine if you are admitted or not. Even Canadian can be rejected at the border if they think your motivation to go to US is suspicious.