All semi autonomous equipment ever since the birth of the computer revolution has had some sort of "Processor" on it, it's just until now it was never really flagged as such.
What your seeing here is the ongoing corruption and half truths that are spread through our society by over zealous marketing agencies, where sales people are encouraged more and more and made to believe they are the stars of the show, simply because they are the one's making the profits.
The fact of the matter is this however, anything that has to perform a set of tasks where the next iteration of a process can be different to the previous iteration, must have some kind of interpreter that can make sense of the instructions the device is given, and then react to those instructions in some fashion.
Back in the mists of time, terminology such as "Controllers" where the norm, but these still boiled down to the same thing.
Take for example an "IDE Hard drive, with it's on board IDE controller", while this is not a CPU in the same sense that you think of a CPU on your PC's main board, it is never the less still a form of CPU.
The host PC sends "OP Codes" (Short for operation codes) across the bus (PCI, ISA, MCI, PCIe or whatever) to the drives controller, the controller then reads this code, and any data that's provided with it and turns them into physical operations that then cause the drive to move the heads to the correct place, and read the requested data.
Routers have an even longer history, Cisco has built networking gear now for best part of the last 50 years or more, and every single one of these devices have had a custom controller/CPU in them all that time. This CPU was designed by Cisco, for Cisco expressly for the purpose of programming and controlling their entire range or Routers & Switches.
Graphics cards are another thing, you hear people band the term "GPU" around like it's some mystical thing that only does graphics. It's not, it's a massively parallel mathematical algorithm processor, Iv'e just finished doing the technical edit on a book on Nvidia CUDA, and what I learned about Nvidia GPU's was rather surprising, these things are Processor's in their own right, processors that are designed to do a specialist set of jobs, but they are still semi intelligent and capable of many different types of operation.
As has been pointed out already, the Netgear Readynas is actually more like a full PC in it's own right. It's just specially designed to function only as a remote storage device.
If you wanted to there would be nothing stopping you from re-programming the Netgear device with new software and making it function perfectly fine as a web server, database server or even a small Linux development server. (A quick search will show you more than a handful of projects aimed to do such a thing with these NAS units)
In terms of processor, well it might surprise you to learn that it's not just Hard drives that have "Processors" on these days, try this little experiment.
Go stand in your kitchen and see just how many CPU's you can count.
I'm willing to bet that your Fridge/Freezer, Washing Machine, Dish washer, Oven and Microwave (at the very least) all have some sort of Processor in, it may not be an Intel Core i7, but it's still a processor, and it's designed to sit their quietly, interpreting instructions sent to it by other electrical/digital circuits which it then turns into the physical operations you see.
So what is the definition of a Processor?
Well it's a bit hard to pin down these days, but in general the definition of a "Processor" is something along the lines of "Any self contained unit, that is capable of acting on external inputs in a semi intelligent way, and producing a known set of outputs derived from those inputs"
SO any stand alone unit, circuit, chip or autonomous machine that can effect a physical manifestation of some known process based on a set of pre-defined inputs can in it's most basic and generic sense be considered to be a processor of some description.