In the Power Transmission world, when we talk about "hardware", we're not talking about bolts, screws, or hinges. "Hardware" refers to the components that the belt runs in. For the most generic name for hardware we will use the name "pulley" - a common name that most will be familiar with.
The terminology for different products is different for different types of belts. Let's look at how the different product types have different specific names for "pulley".
For synchronous belts: "Pulleys" for the current generation of Gates synchronous belts (PowerGrip GT2, Poly Chain GT Carbon) are referred to as a "sprockets". "Sprocket" makes the name association with the ability of current products to replace roller chain - and chain's use of the term "sprocket". Trapezoidal timing belts - the earliest form of synchronous belts, commonly use the term "pulley" to identify their hardware.
For V-belts: "Pulleys" for V-belts are referred to as "sheaves" (pronounced with the "ea" sounding like the "i" in ship, rather than like the "ee" in sheep).
For Micro-V or flat belts: "Pulleys" are pulleys!
So - there are really only a couple of exceptions to remember: "Sprocket" is used for Power Grip GT2 and Poly Chain GT Carbon belt product lines. "Sheave" is used for V-belt product lines. For everything else, you're free to call them "pulleys"!