I think that the term "degree of freedom" had been initially derived from the observation that the system may be completely described by $n$ independent coordinates - they said: "well, if this $n$ is really a characteristic of the system - let's give it a name!". It is my personal belief that the name comes from a classical kinematics: if a body can't move "freely" in some direction - this reduces $n$ by one.
Therefore, your question may be reduced to the following one: "why the number of independent coordinates which describe the system completely does not depend on coordinates themselves?". Am I right here?
Mathematical answer:
We describe systems by means of linear algebra. There is a concept of a basis in linear algebra: a (minimal) set of independent vectors is a basis, while the projection of any vector from a state-space of a system on this set preserves the norm of the state-space vectors (there are many alternative definitions).
What the above definition says is that if you find a set of (independent) vectors, and can represent any state of the system in terms of these vectors without loosing any information - this set of vectors is a basis.
Why do we require the set of vectors to be minimal? Well, you can add another independent vector to this set, but if the system has no component in this additional coordinate - this vector is redundant. Real life example: you can describe the position of a body using three vector coordinates (x, y, z). You can expand this set of three vectors with a vector of velocity of the wind (which is clearly independent with any of the positional vectors of the body), but we need not worry what the wind is. We are measuring the position of a body.
So, we have a basis, what now? Well, there is a theorem in linear algebra which proves that if you have a basis for the state-space, and you find another basis, then the dimensions of these bases are the same. This means that you always need exactly $n$ independent coordinates to describe the system completely (without loss of information).
Intuitive answer:
This one is very tricky, because all the discussion about "coordinates" and "independence" is bound to the framework of linear algebra. Really, I find it very hard to reason the above statements intuitively.
Maybe this: assume that there is finite number of properties of interest in the system (as it is usually the case). You are given $n$ boundary conditions which are not related - none of them can be derived from the rest. In this case, each boundary condition allows you to express a single property of your system. If you find that these $n$ boundary conditions describe all the properties of interest - this means that you have $n$ properties of interest.
Now, if you're given with $m > n$ unrelated boundary conditions, you can express $m$ properties of your system. However, we have already shown that you have just $n$ properties of interest. Conclusion: the $m$ unrelated boundary conditions describe $m - n$ parameters which you are not interested in. Throw them away because they obscure your view.
The answer is two, as can be seen by considering the angular positions of the pulleys: each pulley can be set independently.
I think your error is that the length of rope of the "lone" mass (call it mass number 1) is not a constraint. The position of mass 1 can be set independently of masses 2 and 3. On the other hand, the position of mass 2 cannot be set independently of mass 3, and vice versa.
Best Answer
If you have a system of $N$ particles with $f$ equations of constraints, then the effective degrees of freedom reduces from $3N$ to $3N-f$. This means you don't have to worry about all the $3N$ coordinates, but just focus on the $3N-f$ coordinates to study the dynamics of the system. This idea is based on a simple logic as I will try to explain it in terms of a circular motion.
Suppose a free particle exhibits a circular motion in the $xy$-plane. The solution for the equation of motion will be in the form
$$z(t)=0$$ $$x(t)^2+y(t)^2=r^2$$
where $x(t)$, $y(t)$ and $z(t)$ are the $x$, $y$ and $z$ coordinates at time $t$ and $r$ is the radius of the circular path and is assumed to be a constant. At first sight, a free particle has $3$ degrees of freedom. But the above equation of constraint($x(t)^2+y(t)^2=r^2$) and the restriction of the particle's motion confined to a plane ($z=0$) reduces the degree of freedom. Hence there is in effect $3-2=1$ degree of freedom of the system.
This means you only need any of the coordinates- $x(t)$ or $y(t)$- to describe the mechanics of the system. The choice is yours. If you choose $x(t)$ as the independent coordinate, then since $r$ is a constant, once you fix a value of $x(t)$, the value of $y(t)$ get fixed automatically because of the constraint equation. In other words, $y(t)$ depends on $x(t)$ by the above equation.
Now, you may ask that if you change the coordinate system from Cartesian to some other, say Spherical polar coordinates, then will the dof changes? No, it will not. The choice of a coordinate system will not affect the dynamics of the system. The above circular motion in plane polar coordinates can be written as:
Put $x=rcos\theta$ and $y=rsin\theta$ in the previous equation and we get
$$\phi(t)=0$$ $$(rcos\theta)^2+(rsin\theta)^2=r^2$$
Here, we have $r=\text{constant}$ and hence the only variable that changes with time is $\theta(t)$. So there is only one degree of freedom. You choose $\theta(t)$ as your independent coordinate. As you can see, the degree of freedom is still one.
In the case of free fall of a particle, the solution is given by:
$$y(t)=\frac{1}{2}gt^2$$
where $y(t)$ is the position of the object in the $t^{th}$ second. Here, the degree of freedom is one. You only need $y$ to spot the particle at any time $t$.
In the case of a simple Atwood machine, there is only one degree of freedom. Now, you replace one of the masses by another Atwood machine to form a double Atwood machine (or sometimes called compound Atwood machine). Here the system has two degrees of freedom:
1. one is the freedom of mass $1$ (and the attached movable pulley) to move up and down about the fixed pulley, and
2. one is the freedom of mass $2$ (and the attached mass $3$) to move up and down about the movable pulley
How to do this in terms of constraints?
To describe the configuration of the system, we need 3 coordinates each for masses $m_1$, $m_2$ and $m_3$ and another three for the movable pulley. i.e., a total of 12 coordinates. But thanks to the constraints present. There are $10$ constraints present here:
$8$ of which limit the motion of all the coordinates in a single direction ($x$ I'am taking here);
the remaining $2$ are given by
$(x_p+x_1)=l$ and $(x_2-x_p)+(x_3-x_p)=l'$
where $x_p$ and $x_i$ are the vertical positions of the pulley and the masses $m_i$ respectively.
Hence the effective degree of freedom of the system reduces to $2$.
In a simple way, one degree of freedom is that of the mass $m_1$ and the other is that due to the mass $m_2$. As you can see from the figure, their motions are independent to each other. Hence $x$ and $x'$ are the independent coordinates here.