Installing Starcraft 2 on a second computer

So, the brand new laptop I bought 6 months ago is overheating and I need to send it back to Asus. Luckily, I kept my old desktop. I’d like to install SC2 on it, but it tells me “You already have the maximum number of licenses for this game in this region.” Now, maybe I’m confusing it with Steam, but I assumed that I could install SC2 on all of my computers so long as I associated it with my Blizzard account. Isn’t that the point of syncing all my save games online?

Anyhoo, anyone know how I can install SC2 on my backup computer without having to get in touch with Blizzard’s horrendous support?

Your humble and obedient servant,

Never mind. It appears that if I install it, and just log in with my account, it works fine.