Think of the boundary layer as a multi-lane highway with rubber cars which can bump into each other. This highway has a sticky curb on one side and the cars are a bit sticky themselves, so cars near that curb get the slower the nearer they are.
In one case the cars stay in their lanes and the rightmost lane, right next to the curb, (sorry, you Australians, Japanese or Indians: For you that would be the leftmost lane) is occupied by the slowest vehicles. Speed increases with each lane more distant from this slowest lane since cars rub along nicely. This is like laminar flow.
Now traffic changes and the drivers switch lanes frequently. The result is that cars in the slowest lanes have to speed up. New lanes join the fastest lane from time to time so the speed in the fastest lane will not slow down. Speed is now much more equal across lanes but the whole highway grows wider to accommodate all those new lanes with fast vehicles. This is like turbulent flow.
While in laminar flow the parcels of air all flow in the predominant flow direction, in turbulent flow there is a lot of crossflow, so those parcels get bumped along if friction with the wall (the sticky curb of the highway, to stay in the picture) slows them down too much. This needs a constant addition of new, high-energy parcels so the whole boundary layer is thicker and has a fuller speed profile.
However, if the speed gradient along the predominant flow direction is negative (say, in the recompression area in the rear upper half of an airfoil), the cars in the joining lanes become slower and the slower lanes slow down, too. It's as if they obey a sequence of speed limits that tell everyone to reduce their speed by some MPH. And then some more. If the speed near the curb (in the slowest lane) drops to zero and then reverses, flow separation has occurred. Now the slowest lane fills up with vehicles from both directions which pushes the cars in the adjoining lanes further out. The highway width explodes.
This can both happen with no or much lane changing; the result is the same. When it happens with no lane changing and drivers change their mind about that detail further downstream, the new cars joining will now bump all others along and get traffic moving again. This describes a laminar separation bubble with reattachment downstream.
I am wondering if separation involves only a disturbed boundary layer, while turbulence can involve a wider disturbance such as in a stall?
Every flow separates at the trailing edge. With too much angle of attack, this separation creeps forward on the upper side on thick airfoils or a new separation starts past the suction peak near the nose on thin airfoils. This separation, when extensive enough, causes loss of lift and defines the stall. Both laminar and boundary layers can experience this.
A special case is a laminar separation bubble which occurs past the suction peak but the subsequent transition to turbulent flow causes reattachment. This can still be followed by a separation of the turbulent boundary layer later on.
For example, is it correct to say that in the stall, an already-turbulent flow (sometimes experienced as burbling) becomes detached?
Yes, but also a laminar boundary layer can separate and cause stall (mostly at model airplane scales and smaller). The "burbling" you mention is not caused by this but by larger eddies hitting the tail. This indicates a major separation near the trailing edge on the inner wing but with no or little loss of lift. This kind of turbulence is different from that in a boundary layer and of a much larger scale.
Or that vortex generators, designed to re-energize a stagnating boundary layer, do so by creating turbulence in order to prevent separation?
Yes. Vortex generators add more high-speed lanes to the traffic in the boundary layer. They also help to fix the location of shocks in transsonic flight.