There are no two different categories of limited-slip devices like you are implying. A full, manually controlled, locker rear end (like what is seen in 4-wheel drive vehicles) is extremely uncommon in street/strip applications, and honestly, not really practical unless you are trying to climb out of a swamp and over a dead tree trunk in your Mustang (note my sarcasm). The stock Mustang GT unit is what Ford calls the "Traction Lok", and is simply a multi-clutch disk LSD (limited slip differential). Like said above, the fact that you have a LSD in your car is what makes both wheels spin, and the fact that the 302 is pretty torquey in a small vehicle is what makes it easy to get sideways.
Here is a breakdown of common types of rear ends in street cars:
Open: No clutches, axles are free to spin at any speed in relation to the other. Good for granny driving to the grocery store in her Buick.
LSD: Some amount or type of clutches (though some fancy designs don't have clutches, IIRC) that work to keep the axles spinning at the same or close to the same speeds. These are ideal for accelerating traction, and the Traction Lok that came stock in the 5.0 Mustang is a type of LSD. These are also good for road racing, as their flexibility allows for solid traction in turns without fully locking the axles together.
Spool: A spool is a solid, clutchless, mechanical linkage between the axles in the rear end so that they are always forced to spin at the same rate. Great for drag racing and driving in a straight line, not so great for cornering or trying to navigate a parking lot. They are called "spools" because they kind of look like a thread or rope spool. Typically only used in extreme straight line performance vehicles.
It all comes down to this: it sounds like your LSD is operating properly and the only thing you SHOULD do is buy better all-weather tires and learn to drive more carefully if you intend to drive it in the rain. Your only other option would be to swap in an open differential (which has zero axle locking ability), which is counterproductive from a performance standpoint, and I would NOT recommend it.