Recall that the columns of $A$ span $\mathbb{R}^m$ iff the equation $Ax=b$ has a solution for each $b$. We can restate this fact as follows:
The column space of an $m \times n$ matrix $A$ is all of $\mathbb{R}^m$ if and only if the equation $Ax = b$ has a solution for each $b$ in $\mathbb{R}^m$.
Would the meaning be changed if the "in $\mathbb{R}^m$" was removed in the definition for column space? I'm unsure of why that addition is necessary in the definition, since it wasn't needed in the definition before it.
There is no difference in meaning here: "for each $b$" in statement (1) and "for each $b$ in $\mathbb{R}^m$" in statement (2) mean the same thing. Really both of them should say "for each $b$ in $\mathbb{R}^m$", since taken literally "for each $b$" means for every $b$ no matter what kind of object $b$ is. But this is obviously silly, and in context it is usually fine to say just "for each $b$" meaning "for each $b$ for which the statement would even make sense".