How to understand $df(x,y,z)$? As i have understood it $$\ df(x,y,z)=\lim_{dx,dy,dz\to0}f(x+dx,y,z)-f(x,y,z)\\+f(x,y+dy,z)-f(x,y,z)\\+f(x,y,z+dz)-f(x,y,z)$$
is correct and not$$df(x,y,z)=\lim_{dx,dy,dz\to0}f(x+dx,y+dy,z+dz)-f(x,y,z)$$
But i haven't really understood why it is so. How to understand the meaning of $df(x,y,z)$ ?
 A: To make it more clear, take just the case 2D, and write
$$
\eqalign{
  & f(x + \Delta x,y + \Delta y) - f(x,y) =   \cr 
  &  = f(x + \Delta x,y + \Delta y) - f(x,y + \Delta y)  \cr 
  &  + f(x,y + \Delta y) - f(x,y) =   \cr 
  &  = \left( {{{f(x + \Delta x,y + \Delta y) - f(x,y + \Delta y)} \over {\Delta x}}} \right)\Delta x +   \cr 
  &  + \left( {{{f(x,y + \Delta y) - f(x,y)} \over {\Delta y}}} \right)\Delta y \cr} 
$$
After that, with due considerations about continuity, existence of the limit(s), and of the fact that you get the same result when in the above you invert the path in $\Delta x, \, \Delta y$ (exact differential) etc. , you can conclude that the two definitions are the same.   
It is worthy to mention that, in fact, in the field of Finite Differences, the "offset" in one of the partial difference remains.
The same mechanism extends to 3D.
A: When $f$ is continuous at the point $(x,y,z)$ both limits you have written out have the value $0$; hence they cannot represent the mathematical object $df(x,y,z)$.
Fix a point $p=(p_1,p_2,p_3)$ in the domain of $f$. If $f$ is differentiable at $p$ then
$$f(p+\Delta x)-f(p)=\nabla f(p)\cdot\Delta x+ o\bigl(|\Delta x|)\qquad(\Delta x\to0)\ ,$$
or in coordinates
$$\eqalign{f(p_1+\Delta x_1,p_2&+\Delta x_2,p_3+\Delta x_3)-f(p_1,p_2,p_3)\cr  &=f_{.1}(p)\Delta x_1+f_{.2}(p)\Delta x_2+f_{.3}(p)\Delta x_3 +o\bigl(|\Delta x|\bigr)\qquad(\Delta x\to0)\ .\cr}$$
On the RHS we have a main part which is a linear function of the increment variable $\Delta x$. This linear function is what people call $df(p)$. This means that
$$f(p+\Delta x)-f(p)=df(p).\Delta x \ +o\bigl(|\Delta x|)\qquad(\Delta x\to0)$$
(the . here only denotes application of the linear function $df(p)$ to the increment vector $\Delta x$).
