"Live axle" is a long dead term. People make stuff up just so they can keep using it. Nothing has a "live axle" anymore. All it meant during the horse-drawn days was that instead of having the axle fixed to the springs and the bearings in the wheels, the bearing was on the springs and the wheels were fixed to the axle. In either case, the axle holds the weight. When they started figuring out how to put motors on buggies, they could add a sprocket to a "live axle" and drive it--like a go cart. You can add a differential to this as well--like an ATV uses. The other method, using a "dead axle" was to put a driven sprocket on each wheel and chain drive each wheel separately with a differential between the drive sprockets. The advantage of this is that the axle doesn't have to deal with both the load of the vehicle and the twisting force of the drive power.
This term is long dead and only means anything if you have a go-kart ot ATV where you can choose between one-wheel drive with the sprocket mounted to the wheel or a "live axle" with both wheels driven by a sprocket on the common shaft (differential optional).
What you have now is a beam axle (straight axle) and it's either semi float, with the weight riding on the shafts or full float with the weight riding on a spindle and the drive-axle passing through the center of the spindle and only dealing with twisting forces.
None of that has anything to do with the type of traction device you have. If you welded your spiders together then you made what is know as a spool, essentially. Then you have automatic lockers which use the ground force from tires being pulled along at different speeds to open a dog clutch and release an axle so it can spin independently from it's opposite. Then you have a bunch of limited slip devices that use various technologies to encourage a true differential to give some torque back to that spinning tire. Clutches, viscous fluids with blades chapping away in them and then the Torsen, which is a true differential that uses very inefficient gears to perform its job, which means a lot of the torque which should have gone toward spinning the heck out of the slipping tire is applied to the tire with traction. They paint it up as a brilliant device, but all it amounts to is a brilliant use of a mis-applied gear type.
Then you have the selectable axles where you press a button and an air or electrically powered clutch locks the back up. Some factory vehicles have that--and often times it is computer controlled with a manual over-ride, as is the case with Honda's VTM4 in the Pilot and Ridgeline and others.
In my opinion, the automatic locker is the best for offroad--whether it is a lunchbox locker which uses your factory carrier or a model which replaces your carrier and requires gear set-up. In a medium or heavy truck, and in a 14-bolt GM axle, a full Detroit Locker is a lunchbox locker. In everything else, you have to replace the carrier. I would recommend a Powertrax Lock-rite. I've had one in both ends of my Ranger axles for 15 years. After a week, its quirks and things go away because you get used to them. The awesomeness of never having to say you're sorry never goes away. All wheels drive the same when you need them. Take it easy on slippery side slopes and in 4wd on icy roads.