Why can't polynomials have negative exponents or division by a variable Why can't:

$$2x^{-3} - 3x$$

or

$$\frac{1}{2x}$$

be polynomials too?

Why have a definition that excludes these algebraic forms?
Polynomials are defined as they are for a few distinct reasons: (1) because polynomials as functions have certain properties that your 'polynomials with division' don't have, and (2) because there are other terms for more generalized algebraic forms.

First, the properties of polynomials: unlike e.g., $2x^{-3}+3x$, polynomials have no poles; there's no place where a polynomial 'blows up', where it goes to infinity.  In particular, this means that the domain of a polynomial as a function is all of $\mathbb{R}$ &mdash; or, alternately, all of $\mathbb{C}$).  In fact, polynomials in the complex plane have even nicer properties - they're analytic functions, which means that not only are they well-defined everywhere, but all of their derivatives are well-defined everywhere.  This means that polynomials have a lot of structural properties that make them 'nice' objects of study in ways that your expressions aren't.

Secondly, there's also a notion that (roughly) corresponds to your 'extended' version of polynomials, with its own set of nice properties: the ring of rational functions, which includes not just items like $x^2+\frac1x+5$ but also terms like $\dfrac{x^3+2}{x+5}$ that your formulation doesn't at least at first glance suggest.  These represent a useful extension of polynomials for study because they're closed under the operation of division; for any two rational functions $f(x)$ and $g(x)$, the function $h(x) = \frac{f(x)}{g(x)}$ is also a rational function.  This property doesn't hold for your 'Lambert polynomials', because there's no finite expression in positive and/or negative powers of $x$ that corresponds to the function $\frac1{x+1}$.

What's more, there's also an object occasionally studied that more directly corresponds to your notion: the notion of Laurent Polynomial.  As that article suggests, they're of particular importance and interest for their connections with the field of Hopf Algebras (and by extension, quantum groups).  I would opine that the reason that they're not the primary object of study is because they're not the 'simplest' structure of interest among any of their peers, and fundamentally the most important structures in mathematics tend to be the simplest structures exhibiting some given property.
I would have added this as a comment to Newb's answer but for some reason I can't add a comment today. But in answer to Quora Fea's question in the comments of that answer - is it possibly because a 'polynomial' with negative exponents could be converted into a valid one with positive exponents by multiplying by some x^n. The roots would be the same (with the exception of an extra root at x=0 in the converted polynomial).