TL;DR: It Depends on Purpose.
At the base is a profound mixup of Processor and Processing and both being meant when talking about a CP(U).
Building up thereon
- A Processor is a component in a computer (system). It's usually operational on its own but not necessary useful as such.
- A Computer is a configuration of devices, including a Processor, into a usable system.
The differentiation between either of those can be blurry and depend a lot on time, system type, technology and manufacturer. Not to mention that designers, marketing and especially the general public tend to use terms with way greater freedom (*1).
The Good Old Days.
It was way more clear back then. Take for example a classic IBM 7030 computer. Eventually the fastest scientific computer in 1961, seen here:
(Picture taken from the 7030 Reference Manual p.4)
Except that the number only names the CPU cabinet(s), marked here in red. No Memory, IO or whatsoever - even things we call a bus today was separate, nicely seen on this schematic:
(Picture taken from the 7030 Reference Manual p.6)
(Almost) each of the boxes shown is housed in a dedicated cabinet and connected by arm sized cables. Each of those cabinets was a product on its own, with a product name and to be ordered separate. Memory for example, marked a '1', was known as IBM 7032 Each of those cabinets providing 16 KiWords of RAM.
Same goes for disk controllers, disk drives, tape controllers, tape drives and so on. All one or more dedicated boxes - heck, even the interface for that table sized console needed a cabinet of its own.
Due this quite distinguishable structure the meaning is rather synonymous: The CPU is the Processor and the Processing Unit.
Progress
With time passing units became smaller and higher integrated. Less than 5 years after the 7030 the System /360 was introduced, now components like CPU, Bus controller, I/O Processor and especially Memory became integrated into a single unit - although still occupying several cabinets, depending on machine size and configuration. Only communication controllers,I/O controllers and peripheral devices were still separate units.
While technician still saw and handled the main unit as multiple components (CPU/IOC/Memory Interface/Memory Controller/Memory), most people would call that big box in the middle just CPU.
Mainframe development does continue until today and while some old interfaces have been removed and communication interfaces have been added to the cabinet, the storage is still an external component.
With the integration of multiple processors into a single box a hard dual use of CPU became common: The cabinet at whole being called a CPU, but als talked about multiple CPU being inside.
To solve this IBM started labelling the big box a 'Processor' while the insides doing the computing were called 'Processing Unit(s)'.
Revolt of the Pygmies
About the same time as mainframes started a race to become ever more powerful while even shrinking a bit, other companies focused on a new area, computer systems not as powerful as mainframes, but so small they could fully fit into a few or only a single cabinet. To reach that the processor had to shrink first. DEC was one of them and the PDP-8 eventually their first mini computer. While the first PDP-8 (*2), still filled almost a whole Rack just for the processor (20 HE), the follow up PDP-8/s got it down to a 6 HE box (*3).
With that size (*4) a full computer system, that is CPU with memory, I/O system, console interface, disk(s), (paper) tape and other, could now be as compact as a half height rack.
(Pictures taken from Wikipedia and van der Mark's great PDP-8/E site)
The lower half nicely shows the issue about how to define the CPU part of a processor. In DEC documentation only the 5 left most boards are considered the (KK8E) CPU, but the next two boards hole memory interface, management and sharing, items most would consider part of a CPU - likewise memory control further down.
In the end every manufacturer draws that line different, thus any naming below processor as a blox will be inconsistent.
A 360 Degree Circle ...
Not much later it became possible to integrate a very basic CPU (remember that red line?), now called 'micro-processor', still needing many external components to even start to function. But by now modern PC have taken the full circle back to what mainframes and minis have done with their CPU's.
Just look at one of today's highly integrated microprocessors like AMD Ryzen or Intel Core. They integrate CPU die(s), I/O die(s), memory controller die(s) and maybe cache die(s)in a single multi chip package. It interfaces to the outside only via memory interface, PCIe and USB. Put one on a more basic mainboard that does not add any further peripherals and the result would be quite like what a one would call a CPU back in mainframe days.
... Back to the Naming Issue
With just a single processor in a chip and only one of them in a computer the term CPU could be seen as a distinct naming of a chip and it's solitary nature. People that haven't experienced he development prior would assume that as the only and fixed meaning.
Having two in a computer didn't change much, one calling that a Dual-Processor (or Dual-CPU) sounded great. It became more blurry when they started to integrate two processors into a single chip and a complete mess when systems were created using multiple chips each with multiple processing units
Does It Compute?
The central point to distinguish both might be it's usefulness. A CPU in itself may be operational, but not really useful. Imagine an IBM /360 or DEC PDP-8. Both can run without any peripherals, they can even be programmed using the blinkenlights. Not very comfy. Only by adding peripheral system, most important storage systems a useful computer system is build - with that CPU as it's core component.
It's the very same with a 1960 Mainframe, 1970 Mini or 2020 Desktop.
*1 - Also there's the usual caveat when it comes to any overlapping terminology: In real world communication people care only to differentiate if that is neccessary - otherwise it's an anything goes.
*2 - Introduced 1965, the very same year as the /360
*3 - And cut price almost in half - at cost of reducing speed to ~1/8th.
*5 - Like similar computers of the time and thruout the 70s.
