What a shock absorber actually does
A shock absorber is the damper that forces oil through valves and turns spring motion into heat. It is not the coil spring, not an air bag, and not a MacPherson strut, though that strut uses a damper tube as its locating member.
The spring holds the car up. A seam in the pavement loads that spring, and a spring alone gives the load back as bounce. The part that stops the bounce is the shock absorber. It is a damper. Oil has to pass a valve, and the resistance turns the motion into heat. The tire settles. The coil did not do that.
The drawing is the two housings this damper comes in. Left is the monotube: one cylinder, a piston in oil, a floating piston below it with gas on the far side. The rod has to enter the tube. Oil does not compress, so the gas takes that volume. Valves in the piston meter the oil on the way in and on the way out. That metering is the damping. Right is the twin-tube: a working cylinder inside a reserve tube. The piston valves meter the motion. A foot valve at the bottom sends the oil displaced by the rod into the outer tube, which is the reservoir. Same job, two walls. The gas in a monotube pushes the rod out a little. It does not hold the car up.
On this Fuga the coil is wrapped around the damper. The wheel is off, the disc and the caliper sit beside the stack, and the whole stack gets called a shock, or a strut. Weight sits on the coil. The tube inside the coil is the damper. This is the right front corner of a 2008 Nissan Fuga 350GT, a double-wishbone layout. The arms locate the hub. A tie rod steers it. The damper does neither. A MacPherson strut is the other layout. There the damper tube is the locating member and the steering pivot, and the coil sits on that same tower. A strut cartridge is the damper insert that slides into the housing. The cartridge is a shock. The pivot and the coil are the rest of the strut.
Off the car the split is the same. An eye at each end bolts the damper to the suspension and the body. The coil around the tube can come off. The tube that is left is the shock. An air bag is a spring that carries the car on air pressure instead of a coil of steel. It is not a damper. The unit sold as an air shock puts a bag and a damper together. The rubber holds the weight. The oil tube still kills the oscillation.
A worn damper lets the spring finish its own argument. The nose floats after a dip, or the tire pats the road in a short hop. The coil can be the right rate and still do that, once the valves in the tube have quit.