I got a confirmation that my DS-2019 form is on the way, so I can soon request my J-1 visa. This form basically confirms that Stanford will accept me as an exchange scholar, and so I can request an exchange scholar visa.
The process is similar to requesting a visitor visa (B1/B2), but requires a few extra step, like paying a SEVIS fee. That fee goes to support the US government spying on me and making sure I stay in school and don't, say, go and blow up the world trade center. Yet another tax that did not exist before 9/11.