1-dim representations of the affine Hecke algebra for $G = \text{SL}_2$

The discrepancy comes from the fact that there are two different affine Hecke algebras, one for the weight lattice and one for the root lattice. The root lattice version is a subalgebra of the weight lattice version. Concretely, the root lattice version is the one appearing in your method 1. It may be realised as the quotient of the group algebra of the free group on two generators $T_0$ and $T_1$ modulo the quadratic relations $$(T_i+1)(T_i-q)=0 \ \text{for} \ i=0,1,$$ explaining the appearance of the four one-dimensional representations: each generator has two possible eigenvalues, which may be chosen independently. The weight lattice version is obtained from this by adjoining the outer automorphism interchanging the two generators. It has only two $1$-dimensional representations because $T_0$ and $T_1$ must now have the same eigenvalues.

Geometrically, the difference is that one of these comes from $\text{SL}_2$ and one from $\text{PSL}_2$.

I think I know what's wrong. It turns out I was confused about the statement on the geometric side. In the Ginzburg/Chriss text, the irreducible representations are not exactly the Borel-Moore homology of the fibers but the image of Borel-Moore homology of some tubular neighborhood. If one takes such a neighborhood the two-dimensional representation I referenced above should be the Borel-Moore homology of a point union a small disk in $\mathbb{A}^1$. However, Borel-Moore homology of open disks vanishes, so the representation I claimed was two-dimensional is really one-dimensional.

Perhaps more convincingly, one can refer to paper as a reference. One doesn't just take the (co)homology of the fibers for each orbit representative. The statement (page 44) is really that the irreducibles (for a given central character) are the summands we get by applying the decomposition to the map $\mu$: $$\mu_* \mathbb{C}_{\tilde{\mathcal{N}}^a} = \bigoplus_{\phi} L_\phi(k) \otimes IC(k)$$

In our case, the map is $\mu: \mathbb{A}^1 \cup \text{pt} \rightarrow \mathbb{A}^1$. The cohomology upstairs is $\mathbb{C}^2$, and decomposes into two one-dimensional representations. So for each central character has two one-dimensional representations as expected; one for each $C(g) = \mathbb{G}_m$ orbit in $\mathbb{A}^1$.

The following was useful for me, just to verify computations if anything: https://www.math.hmc.edu/~davis/ThesisFinal.pdf Also, thanks to the comments and answers, they were useful and informative nonetheless!