I think a driver is, sort of, part of a module. Is that right?

As I understand it, a module is a kernel plug-in. It's code that runs in "kernel space", as if it were built into the kernel. One of the uses of kernel modules is to provide hardware drivers, but that's not the only thing they can provide.

A driver is simply a bit of code that tells the computer how to interface with a certain hardware device. In linux, they are most commonly designed as kernel modules (probably for efficiency), but it is possible to create module that work outside of it, in "user space". The ntfs-3g driver is an example of a non-module driver.

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Funny, that FS driver is an example of driver that does not communicate directly with hardware... So your example shows not the point you wanted to illustrate. It is easy to give an example of device driver being not entirely in kernel space: wireless drivers load firmware with help from userspace applications which find firmware and copy it to a special place on /sys filesystem.

From what I understand, it goes like this:

Dog is to Animal
-As-
Driver is to Module

Not all animals are dogs, but all dogs are animals.
Not all modules are drivers, but all drivers are modules.

A driver is like a sub-class of module.

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As far as I understand, video drivers in X are working one half in userspace and one half in kernel space. So each driver contains a module, but may be bigger (on the other hand, the module can be not exactly module when it cannot be omitted on any architecture where it can be built at all).