I am a U.S. citizen living in Russia with a multi-entry student visa. I noticed that on my migration card it lists the Russian Federation as well as the Republic of Belarus. Do I need to get another visa if I want to visit Belarus, or is what I already have sufficient?
New Russia-Belarus visa agreement (2020-07-01)
As of 2020-07-01, you no longer need a separate visa, as Russia and Belarus recognise each others visas. See news articles from Belarus Feed, Belarusian Telegraph Agency, TASS, and intourist. See also this answer about crossing the Russia-Belarus border.
Warning as of 2020-07-07: another source in Russian, dated 2020-07-01, states it will take several more months until it actually takes force. These sources contradict each other. Either it's already in force, or it will be soon (and as of July 2020, the 2019-2021 Coronavirus Pandemic means few travellers should make the overland EU-Belarus-Russia trip anyway).
In short the answer is yes you do need a separate visa for Belarus. Your Russian Student Visa doesn't allow you to enter Belarus.
The reason that it has the names of both countries is because the form they use is exactly the same. From the US Embassy in Moscow:
Although Russia and Belarus use the same migration card, travelers should be aware that each country maintains its own visa regime. U.S. citizens wishing to travel to both nations must apply for two separate visas. A traveler entering Russia directly from Belarus is not required to obtain a new migration card, but at his or her option may do so if blank ones are available at the time of entry.
Definitely do need a Belarus visa even if there is no passport control between Russia and Belarus. Flights between Belarus and Russia are domestic with no passport control. Random document checks do happen by Belarusian authority on the train and airport as it happened to me once at the airport. Also, the police can stop you to check your documents. You don't want to be caught without a visa and it's not difficult to get a visa, too.
Similar to the Schengen Zone, Russia and Belarus do not have passport control between the two countries.
With that said, http://mfa.gov.by/en/visa/ doesn't appear to explicitly list any exception for travel from Russia, so, technically, you might still need to have a visa in case you're subject to a random check.