How can friction be the driving force of a car, while at the same time slowing it down? Is was reading an explanation of how friction makes a car move which stated that the static friction between the road and the car 'locks' one part of the tyre in place and that, because the wheel still rotates, this causes a forward motion. 
Why then, is friction between the tyres and the road often giving as a force that slows the car down?
My hypothesis is that the 'slowing friction' is a different thing than the 'pushing friction', but the two confusingly share the same name. More specifically, I think the 'slowing friction' is the rolling resistance of the tyres, which (I think) is an effect of the material properties of the tyre. Is this correct? If not, than what is the difference?
 A: Forget about pushing friction and slowing friction. Think of static friction and kinetic friction. 
Static friction is friction between two or more solid objects that are not moving relative to each other. It's what keeps the car from slipping. When the car is in motion, ideally, the tyre and road surface do not move with respect to one another, the tyre grabs the road. It works the same way with the soles of your shoes and ground.
Work done is equal to Force times Distance. Since the tyre and road are not moving with respect to each other, no work is done against static friction, nor can it ever. When you are cruising at a constant speed on a level road, the engine is working against friction, but this is kinetic friction: the friction between internal parts of the car's engine and drivetrain and the friction between the car's body and the air.   
There is also loss within the tyre as it rotates. The tyre flexes as different sections of the tyre come into contact with the road during rotation. The deformation is not perfectly elastic and some of the energy is lost as heat during the process. Underinflated tyres can add to the effect and increase fuel consumption. Recommended tyre pressure is a trade off between comfort and handling. 
When you apply the brakes, they are designed to cause kinetic friction between the brake components (pads and rotors for disk brakes, shoes and drums for conventional) which converts the kinetic energy of the car into heat. Electrical cars can convert some of the kinetic energy back to electrical energy which is a more efficient use of the kinetic energy. 
When the tyres slide, as when you go into a skid, kinetic friction between the tyre and the road does slow the car down, but nowhere nearly as efficiently as the brakes would, which is why modern cars have anti-lock brakes. Besides, steering is nil when in a skid. 
A: Your hypothesis is a much better understanding than that given in your second link.
Friction, in a literal sense, is the source of all of a car's acceleration which is not being caused by aerodynamic forces and gravity.  Thus, on level ground in a vacuum and at constant speed, the horizontal frictional force will be zero.
In this constant speed scenario, the engine is still doing work, and the tires are still dissipating much of that energy through contact forces with the ground.  You might wonder at this point how the contact force between the tire and ground exerts torque on the axle if the horizontal frictional force is zero.
This is possible because the tire contacts the ground not at a point but over an area which extends slightly ahead of and behind the axle.  While the tire is rolling, the normal force is greater in the front half of the contact area than the back half, exerting a net torque about the axle.
So yes, the "rolling friction" of the tires is qualitatively different than the standard frictional force.
