Why does the inner product of two vectors have to be positive definite?

Yes, it is part of the defion of inner product that we always have $\langle v,v\rangle\geqslant0$. That's because that allows us to define a norm $\lVert v\rVert=\sqrt{\langle v,v\rangle}$ and from that norm we get a distance: the distance from $v$ to $w$ is $\lVert v-w\rVert$.

But I don't think I've ever seen “Positive definiteness” as a name for this property. It has nothing to do with positive definite matrices.

Your confusion stems from this:

I merely learned that the inner product of two vectors $\mathbf{a}$ and $\mathbf{b}$ is:

$$\mathbf{a} \cdot \mathbf{b} = \sum_{i = 1}^n a_ib_i$$

This is the usual definition of inner product in $\Bbb R^n$. In more advanced classes, we learn that there are other possible definitions of an inner product on a vector space. But if we want to call $\left<x,y\right>$ an inner product, it has to obey certain conditions, one of which is that $\left<x,x\right>\ge 0$, with $\left<x,x\right>=0$ if and only if $x=0$.

By the way, the definition of positive definiteness that you give in your question is garbled. It should be something like:

Positive definiteness: $\langle\mathbf{a}, \mathbf{a} \rangle \ge 0$ for all $\mathbf{a}$, and the necessary and sufficient condition for $\langle\mathbf{a}, \mathbf{a}\rangle = 0$ is $\mathbf{a} = \mathbf{0}$.