What is the Euler Totient of Zero? Wolfram MathWorld defines the Euler Totient function as follows:
The totient function phi(n), also called Euler's totient function, is defined    
as the number of positive integers <=n that are relatively prime to (i.e., do 
not contain any factor in common with) n, where 1 is counted as being 
relatively prime to all numbers

By this definition would $\phi(0)=0$?

In general, we don't often need to define $\phi(n)$ for $n<1$. 
For $n=1$, we define $\phi(1)=1$ because we want $\phi$ to be multiplicative, and because it lets us deduce that $\sum_{d\mid n} \phi(d) = n$. The whole reason Wolfram defines it with $\leq n$ rather than $<n$ is just for that case of $n=1$. Essentially, if $\rho$ is any multiplicative function, then $\rho(1)$ is a "empty product," and thus needs to be $1$.
There isn't much value in defining $\phi(0)$. Sometimes, we just leave things undefined. In broader contexts, like algebraic number fields, you can define $\phi(I)$ where $I$ is any ideal in a commutative ring, and $\phi(I)$ is the number of units in $R/I$. That yields a definition of $\phi$ that is more natural and you get $\phi(0)=\phi(0\mathbb Z)=2$. But I don't think that value is useful.
