I have a trouble understanding how the MOS 6502 clock works.
Erm, reading through the question it seems you're actually asking why different 6500 CPU variants each need a different clock setup.
Possibly due to my extremely low knowledge regarding electronics in general.
This isn't about 'electronics knowledge' (*1) but simply designated use case for each 6500 variant.
The 6501 ...
... is a straight drop in replacement for the 6800. That means it can be plugged directly into a computer built for a 6800.
The 6800 has no internal clock generator, but needs to be fed two non-overlapping clock signals, such as those created by the Motorola 687x family of integrated clock generators (*2). To make a 6501 work in a computer designed for the 6800 it has to the same pinout and behave exactly like a 6800 regarding their operation, i.e. all signals, which of course includes taking the Phi1/2 clock signals and using them like a 6800 would.
... is a new design, based on the 6800 design, but radically simplified and improved. As such, it relies on the same non-overlapping two-phase clock signal to manage its workings. It has a different pinout than the 6501 and cannot be used with 6800 boards.
One improvement over the 6800 was that it includes an on-board clock generator. Thus all it needs is an external oscillator - which can be as simple as a RC-circuit - this saves quite some money. Motorola priced the 6870 in 1976 at 20-30 USD, depending on volume. That the 6502 cost less than that for the CPU including clock generator does make a point (*3).
The 6512 ...
... in turn is a 6502 regarding the signalling exactly like the 6502, except it needs to be fed a two-phase signal exactly like the 6501(6800). The internal clock generator is disabled.
While many standard applications work quite fine with a simple symmetric single-speed clock, there are cases where it is of advantage to have fine control over how the CPU is clocked in general and at times. For example, working with an asymmetric clock structure with varying clock speed: for synchronisation with external devices, or more generally to fit with external timing of whatever structure.
The eventually most well-known example of a 6512 may be the BBC Micro B+ using it instead of the 6502 the Model A had.(*4)
6501, 6502 and 6512 are three variants of the same CPU for different use cases each requiring a different approach to clock generation
P.S.: That cited site about 6500 clock is not entirely correct when saying
The two clocks are basically inverted versions of each other.
No, they are complementary non overlapping. That means they are not simply inverted, but delayed in a way that no two transitions occur at the same time. This is due to the fact that the two clock signals are needed to produce 4 distinct clock signals used to crank all internal workings.
The 6502 receives an external square-wave clock input signal on pin 37
Again no, it need not be a square wave, only a wave form that stays long enough above and below a certain threshold. It can be a square wave, but does not have to be.
Or in other words, that page is well ... let's say 'oversimplified' without understanding what it tells.
*1 - Maybe a bit, as to the difference between an oscillator and a clock generator:
- An oscillator is some device that delivers a periodic signal, for example a crystal or a RC-circuit, while
- a clock generator takes the periodic signal and turns it into one (or more) clock signal(s) as needed by the CPU.
*2 - A very interesting circuit, as what looks like a strange crystal is in fact a hybrid circuit combining a crystal, clock circuit and drivers in one package.
*3 - So maybe better qualified as clock generator with CPU included for free?
*4 - Often the Apple II is cited as already doing this with a 6502, as Woz created a rather complex clock circuit that fed the 6502 as well. But doing so with Phi0 on a 6502 only allows changing the basic clock frequency and symmetry, not not handling Phi1 and Phi2 separately as the 6512 does.