"When I use a word," Humpty Dumpty said, in a rather a scornful tone, "it means just what I choose it to mean—neither more nor less."
I think Arnol'd is correct, but I think he is being unnecessarily confrontational about it. All the books on your list that I am familiar with nearly immediately jump to a more precise formulation that a differential equation is one of the two following things:
\[
y^{(n)}(t) = F(t, y(t), y'(t), \dots, y^{(n-1)}(t) ),
\]
or
\[
G(t, y(t), \dots, y^{(n)}(t)) = 0.
\]
Here is another example of an equation that I would not want to call a differential equation:
\[
y'(t) = y(t-1).
\]
This meets the heuristic definition, but fails to be of the form I specified above (or of the form Arnol'd considers).
I now see that Qiaochu has written nearly the same thing above.
btw, I think Arnold's book is fantastic, but should be complemented with a more standard treatment of ODE, if only so that you know what everyone else knows in addition to the topics Arnold focuses on.
EDIT: To answer the 2nd half of the question, I don't know of any books that are as geometric as Arnold. IMO, the big strength of his book is that he makes the geometric intuition jump out at the reader, and downplays the analytical side of things. This complements the more traditional books that focus on the analytical aspects (and on explicit solutions) and lose all the geometry.
Arnold has another book that is somewhat more advanced, Mathematical Methods of Classical Mechanics. I think it's another great book, though it's hard to read. He also has a book called Geometrical methods in the theory of ODE. This is also a more advanced book, so it is not one you want to look at yet.
A book that I found very compelling was Hirsch and Smale, Differential Equations, Dynamical Systems and Linear Algebra. It's more analytical than Arnold, but is more geometric than most.
EDIT 8 years later:
Let me add a recommendation for Strogatz's Nonlinear dynamics and chaos. I think it's a beautiful book and wish I could go back in time and give it to my younger self.
A partial differential equation is one for which there are several independent variables. In contrast, an ordinary differential equation has just one independent variable. For example, see this discussion of PDEs or see this discussion of ODEs.
Often, in the standard examples of PDEs we separate solutions of several variables into products of functions of one variable. The one-variable functions are solutions to an ODE which is derived from the given PDE. This technique is known as separation of variables.
Best Answer
An ordinary differential equation is an equation which involves derivatives of one or more dependent variables with respect to a single independent variable.
Ex: $\frac{d^2y}{dx^2}+\frac{dy}{dx}+2=0$
A partial differential equation is an equation which involves partial derivatives of one or more dependent variables with respect to more than one independent variables.
Ex : $u_{xx}+u_{yy}=0$