Two different definitions of ellipticity

The answer is no. For instance, definition 2 can be satisfied by subelliptic operators which are not elliptic in the sense of definition 1. The sublaplacian on the Heisenberg group is an example.

I'll add some more details later when I have more time.

Edit: Actually, unless I am mistaken, even a fully degenerate operator can satisfy definition 2. Let $\Omega = (0,1)^2$ be the open unit square in $\mathbb{R}^2$, and let $L = - \frac{\partial^2} {\partial x^2}$. We know that $-\frac{d^2}{dx^2}$ is elliptic in all senses on $(0,1)$, so if $u \in C^\infty_c(\Omega)$, then for each $y \in (0,1)$ we have $$-\int_0^1 u_{xx}(x,y) u(x,y) dx \ge c \int_0^1 |u(x,y)|^2 dx$$ for a constant $c$ independent of $y$. Integrating with respect to $y$ now gives the result.