What you wrote is correct. Every request for a stay goes first to the Justice assigned to that Circuit; if that Justice denies a stay, a party can, in theory, keep asking one Justice after another until they've asked all 9.
In practice, every time a request is made to a second Justice-- and almost every first request in a high-profile or controversial case-- the Justice who gets the request will ask the full court to decide it (as Sotomayor did here). That is simply to save time, because in those situations it's clear that multiple requests are going to be made.
What I do think is significant is that there was no recorded dissent from the grant of a stay in this case.
Do tell? Is it common to see such dissents or is it that you expected ideological lines to be drawn this early with respect to a stay request for same-sex marriage?