I am a little confused by the definition of ellipticity.
(A) On one hand site, I find the following definition: The operator $Lu(x)=\sum_{i,j=1}^d a_{i,j}(x)u_{x_i}u_{x_j}(x)+...$ is called elliptic in $x$ if the matrix $a(x)=(a_{ij}(x))_{i,j=1,..,d}$ satisfies $\xi^Ta(x)\xi\neq 0$ for all $\mathbb{R}^d\ni\xi\neq 0$.It is called elliptic on a domain $D$ if it is elliptic in each point in $D$.
(B) On the other hand, one often finds the definition that the above matrix $a$ needs to be positive-definite at each $x\in D$, in order for the operator to be called elliptic.
Now, clearly (B) implies (A). For the other direction, I understand that we can assume without loss of generality that the matrix $a(x)$ is symmetric at each fixed $x\in D$. Then, it is clear, that the matrix is either positive-definite or negative-definite at each fixed $x$. However, it can still be negative definite for one $x$ and positive-definite for the other. So how can I assume that it can be positive definite for all $x$ without loss of generality? Please note that I do not want to make the assumption that the coefficient matrix $a$ is continuous.
There is no such thing as "the definition of ellipticity". There are definitions of ellipticity, not mutually compatible, each adapted to a particular context (class of PDE problems). Someone working with constant/smooth coefficients will define it differently from someone working with measurable coefficients, who will define it differently from someone working with nonlinear equations. Do not assume that a definition made in one context (e.g., smooth coefficients) can be applied to another.
For your specific question: the first definition is meant to be used for smooth or at least continuous coefficients; it is not what you want to use when the coefficients are discontinuous.