The construction of a maximal non-split torus has always felt trickier to me than the other two you mention, but it isn't too hard to understand. I learned most of this by reading Cédric Bonnafé's text Representations of $SL_2(\mathbb{F}_q).$

If we take $K / F$ to be a field extension of degree two, then we can regard $K$ as a two dimensional vector space over $F$, and as multiplication by elements of $K$ is an $F$-linear operation, by choosing a basis of $K$ as a vector space over $F$ we get that $K^\times$ is isomorphic to some subgroup of $GL_2(F)$. Through this isomorphism, the trace and norm of the field extension correspond respectively to the trace and determinant of the matrices. This shows that the elements of $K$ which are of norm 1 over $F$ can be identified with a subgroup of $SL_2(F)$, this subgroup is a maximal non-split torus.

For me the problem with this construction is that it doesn't necessarily make it easy to see what the matrices in this non-split torus look like. What helps me is to try and understand them through analogy with $SL_2(\mathbb{R})$, because there I know a field extension of degree 2 that I understand well. Using 1 and $i$ as our basis for $\mathbb{C}$ and tracing through the isomorphisms there gives us that our non-split torus is $SO_2(\mathbb{R})$ as a subgroup of $SL_2(\mathbb{R})$ in the usual way. While I feel I can get a good mental grasp on this group, it still feels quite a bit more complicated to me than the standard split torus.

I hope you find this useful, It is treated fairly well in Bonnafé's aforementioned text, all in chapter 1, section 1.1.2. There are some errors in some later calculations in the text, but the section on the non-split torus is good, and there is a good exercise on calculating a good basis for $\mathbb{F}_{q^2}$ over $\mathbb{F}_q$ for calculating the matrices you get in the non-split torus.