under my ID and redeem the Lion 10.7 that I purchased
Since you already have Lion purchased under your own account, it will always be there for you to retrieve without paying for it again. You are allowed to install Lion on as many Macs as you own, so it doesn't matter if you purchased it for a different Mac. Just login to your account with the Mac you're referring to here and install Lion.
An Internet Recovery will only work if your Mac came with Lion or later. Anything earlier, you have to install from the DVD your Mac came with, or a compatible retail disk. The Mac in question was originally released in May, 2011. According to www.everymac.com, they shipped with Snow Leopard, 10.6.3, or 10.6.6, depending on the model purchased, and when.
Even at 10.6.3, it's questionable a retail copy of Snow Leopard would work. If it came with 10.6.6, it definitely won't. If you didn't get the original gray disks with the Mac, see if you can get those from the seller. You'll need them. If s/he doesn't have them, you'll have to get a replacement set from Apple. They'll need the serial number to make sure you get the correct disks.
Once you have the Snow Leopard disks for your Mac, boot to the DVD, erase the drive and reinstall. Update if necessary to get the App Store on your drive. Then login to your account and reinstall Lion. Though if you want to, you can skip directly to Mavericks.