Why are there so many different definitions for differentiability? I am studying differentiability for functions of several variables.
Here is the first definition of differentiability I came across:$\quad$
A function $f:\Bbb R^n\to\Bbb R^m$ is differentiable at $a\in\Bbb R^n$ if there is a linear transformation $\lambda:\Bbb R^n\to\Bbb R^m$ such that $$\lim_{h\to 0}\frac{\left|\,f(a+h)-f(a)-\lambda(h)\,\right|}{\left|\,h\,\right|}=0.$$
Next I found this other definition:$\quad$A function $f:\Bbb R^n\to\Bbb R^m$ is differentiable at $a\in\Bbb R^n$ if there is a linear transformation $\lambda:\Bbb R^n\to\Bbb R^m$ such that $$\lim_{h\to 0}\frac{f(a+h)-f(a)-\lambda(h)}{\left|\,h\,\right|}=0.$$
Finally I found the following one:$\quad$ A function $f:\Bbb R^n\to\Bbb R^m$ is differentiable at $a\in\Bbb R^n$ if there is a linear transformation $\lambda:\Bbb R^n\to\Bbb R^m$ and a function $r:\Bbb R^n\to\Bbb R^m$ such that $\lim\limits_{h\to 0}\,\,r(h)=0$ and $$f(a+h)-f(a)-\lambda(h)=\left|\,h\right|\,r(h)$$
Are all these definitions equivalent?  Why are there so many definitions for a single concept?  What are the advantages of one definition over the others? 

I disagree that these definitions are different - they are just the same definition written in three different ways.  The first definition is what you will see in most textbooks; I would consider it to be the "standard" definition.  To parse the second definition, recall that a sequence $v_k$ of vectors in Euclidean space converges to $v$ if the sequence $|v_k - v|$ of real numbers converges to $0$ (this is nothing more than the definition of convergence in Euclidean space).
Finally, the third definition should read:
$$f(a + h) - f(a) - \lambda(h) = |h|r(h)$$
for some function $r(h)$ which converges to $0$ as $h \to 0$.  (Actually, some authors will write $\leq$ instead of $<$.)  In any event, you can just solve this equation for $r(h)$ to get:
$$r(h) = \frac{f(a+h) - f(a) - \lambda(h)}{|h|} \to 0$$
which is exactly the second definition.
