"There is absolutely nothing light, about Light Infantry..."
This is truth. Light Infantry just means that you don't have any heavy vehicles to carry your gear for you...
I'd like to think of Light Infantry as "armed with light weaponry" (primarily small arms) as opposed to Heavy Infantry as "armed with heavy weaponry" (such as towed howitzers and such). Vehicular support would make them more akin to Mechanized Infantry.
Plus, a soldier's expected to carry all of your fighting gear while out on patrol. If you can't carry it, you're not gonna be using it anyway.