Hecke algebra of GL(2,F)

A Hecke algebra, in the most common definition, is associated to a pair of groups $G$ and $K$, and is the convolution algebra of bi-$K$-invariant distributions on $G$.

As far as I know the left-invariant distributions are never considered to be a Hecke algebra. This algebra has a highly degenerate structure, because if $a,b$ are distributions and $a'$ is the average of the right translates of $a$ by elements of $K$, then $ab=a'b$, so $(a-a')b$ is zero. So the algebra has a huge space of zero-divisors, and in particular does not have a unit. If you mod out by the degenerate elements $a-a'$, you obtain the usual Hecke algebra of bi-invariant functions.

The definitions you see involving the Weyl group are most likely Iwahori-Hecke algebras, as Marty points out. In this case we would take, if $F =\mathbb Q_p$, $K$ to be the subgroup of elements of $GL_2(\mathbb Z_p)$ that are congruent to upper-triangular matrices mod $p$.

In addition to the link Marty provided, I think your idea of trying to work out at least some of the calculations yourself in the $GL_2$ case is a good one, by using the Bruhat decomposition to find $K \backslash GL_2 (F) / K$ and then computing the products of some double cosets.

Let $G$ be a reductive $p$-adic group. First fixing a Haar measure on $G$, you can identify the algebra of distributions of $G$ with the ("big") Hecke algebra $H(G)$ of locally constant complex functions with compact support equipped with convolution $\star$. If $K$ is any compact open subgroup, the bi-$K$-invariant functions form a subalgebra $H(G,K)$, called the $K$-spherical Hecke algebra. More generally if $\rho$ is an irreducible smooth representation of $K$, you may form the subalgebra $H(G,\rho ) = e_\rho \star H(G)\star e_\rho$, where $e_\rho \in H(G)$ is the idempotent attached to $\rho$. Note that $H(G,K)$ corresponds to the particular case where $e_\rho$ is the trivial character.

All these algebras are called Hecke algebras. On the other hand there are standard Hecke algebras defined in an algebraic manner via generators and relations, like Iwahori-Hecke algebras.

You cannot expect to understand any sort of Hecke algebra in a naive approach. Any progress in the description of these algebras rely on a very fine understanding of the structure of $G$ (in particular of the double coset set $K\backslash G/K$) as well as a deep understanding of the structure of the representations. In general one wants to find an explicit isomorphism between an Hecke algebra that one wants to understand with a standard Hecke algebra.

Historically the first results concerned the spherical Hecke algebra $H(G,K)$, where $K$ is a "special" maximal subgroup of $G$ (because of its importance in the theory of automorphic forms) via the Satake isomorphism and the Iwahori Hecke algebra (when $K$ is an Iwahori subgroup), because of its importance in the study of unramified principal series.

The general understanding of the Hecke algebras that arise from reductive $p$-adic groups is one of the aims of Type Theory as conceptualized by Bushnell and Kutzko. It turns out that a lot of those Hecke algebras are in fact isomorphic to (or at least Morita equivalent to) standard Hecke algebras! But this is not trivial: this is the result of decades of research starting in the $50$'s.