There is a Cartan decomposition for $PGL_2(\mathbb{Q}_p)$ meaning that double coset of the from $PGL_2(\mathbb{Q}_p) //PGL_2(\mathbb{Z}_p)$ are represented by $\alpha^k$, $k\geq0$.

So this is the reason because the definitions are equivalent.

This works well as long as you are working on $GL_2(\mathbb{Z})$ automorphic forms with preassigned behaviour on the center, for congruence subgroups. You need then a more general definition. Also it seems to me that you do not carefully distinguish between the adelic setting and classical setting concerning Lie groups only.

For the general case: The algebra of Hecke operator is $C_c^\infty( G(A_f) //K)$ acting by convolution on $L^2( G(\mathbb{Q}) \backslash G(A) /K_f)$ if $K_f$ is an open  subgroup in $G(A_f)$ and $\Gamma = K_f \cap G(\mathbb{Q})$. ($A=$adeles, $A_f=$ finite adeles). You have on the level of $G(\mathbb{R})$-reps under certain conditions (see strong approximation) that $L^2( G(\mathbb{Q}) \backslash G(A) /K_f) \cong L^2(\Gamma \backslash G(\mathbb{R}))$.