A DVD will always install whatever OS is on it. When you you boot to Internet Recovery Mode, it first looks to the hidden partition on the drive and will install that same version of OS. If that partition is missing (which we already know it is), it drops back to Lion.
And that can depend on the model. We have a Mac Mini that shipped with Snow Leopard that has a firmware update available for it that would add the ability to do an Internet Recovery Mode startup. However, it then also makes it impossible to install Snow Leopard from its own original disks! So that firmware update is NEVER gonna' happen on this Mac since the entire reason for purchasing it was to run PowerPC scanner software that is highly unlikely to ever be updated to an Intel native version.
It would appear then that your Mac has had a firmware update applied to it that allows Internet Startup Mode. Or, since any Mid 2010 models sold in early 2011 were so close to the switch at Apple from Snow Leopard to Lion, it may have been built with the firmware for Lion capability, but shipped with Snow Leopard disks anyway. But that's strictly a guess. Our mid 2010 Mac Pro was purchased early 2011 to get one of the last ones that shipped with Snow Leopard, but it has no Internet Recovery Mode ability.
So I can't quite be sure what's going on with your Mac. That it stopped at about the same point can suggest various problems. One being the DVD has something wrong with it. Two, the optical drive has a problem - but stopping at the same point would be pretty unusual for a hardware issue there. Three, the hard drive may have a faulty block/sector it can't get past. To possibly fix that, you would to a single pass erase on the entire drive with Disk Utility. Any bad blocks will be mapped out. This will take a while.
Any of these may be the issue, or maybe even none of them. They're just some of the more common reasons for any computer.