I would rather see $x$, $y$ and $z$ instead of $x_1$, $x_2$ and $x_3$... But it is still a linear equation because the degree of the polynomial is $1$. I know what you are stuck on, the fact that the coefficients are $0$, therefore making the left-hand-side evaluate to $0$, which is degree $0$. But because of the variables, we have to call it a linear equation. Just think of it as a linear equation with no solutions (because $0$ $\neq$ $5$).
Quadratic equations are of the form $Ax^2 + Bx + C = 0$, where $A$, $B$ and $C$ are all real numbers. The thing different about this equation is that the degree is $2$, not $1$. Similarly, cubic equations have degree $3$, quartic equations have degree $4$, and so on. Equations of degree $0$ have no special name.
Many people get confused when they see things like $x_1$, $x_2$, and $x_3$. Whenever you see something like that, think of it as just being $x$, $y$ and $z$. They are different variables and are in no way related to each other.