Is $0$ an imaginary number?

The Wikipedia article cites a textbook that manages to confuse the issue further:

Purely imaginary (complex) number : A complex number $z = x + iy$ is called a purely imaginary number iff $x=0$ i.e. $R(z) = 0$.

Imaginary number : A complex number $z = x + iy$ is said to be an imaginary number if and only if $y \ne 0$ i.e., $I(z) \ne 0$.

This is a slightly different usage of the word "imaginary", meaning "non-real": among the complex numbers, those that aren't real we call imaginary, and a further subset of those (with real part $0$) are purely imaginary. Except that by this definition, $0$ is clearly purely imaginary but not imaginary!

Anyway, anybody can write a textbook, so I think that the real test is this: does $0$ have the properties we want a (purely) imaginary number to have?

I can't (and MSE can't) think of any useful properties of purely imaginary complex numbers $z$ apart from the characterization that $|e^{z}| = 1$. But $0$ clearly has this property, so we should consider it purely imaginary.

(On the other hand, $0$ has all of the properties a real number should have, being real; so it makes some amount of sense to also say that it's purely imaginary but not imaginary at the same time.)

I don't think there is a

complete and formal definition of "imaginary number"

It's a useful term sometimes. It's an author's responsibility to make clear what he or she means in any particular context where precision matters. If $0$ should count, or not, then the text must say so.

Your question shows clearly that you understand the structure of the complex numbers, so you should be able to make sense of any passage you encounter.