Also, it's not clear why he would have wanted to stay in Moscow if he had a choice. If he had prior contact with Russian authorities, they presumably wouldn't have made him stay at Sheremetyevo for over a month.
Also it's worth noting that in 2013 Russia had much friendlier relations with the US. This was pre-Crimea. It’s not surprising that it took Russia so long to decide what to do with Snowden. Eventually, they concluded that letting him stay wouldn’t significantly harm their relationship with the US while also gaining goodwill elsewhere.