Rank of a matrix

Question

Let $A$ be an $n \times m$ matrix.

1. If $Ax = b$ has a solution for every $b$, what is the rank of $A$?
2. If $Ax = b$ has a unique solution, what is the rank of $A$?

My attempt:

1. $Ax = b$ has a solution for every $b$ implies that there is a pivot in every row. Therefore, $rank(A) = m$.
2. $Ax = b$ has a unique solution means that there is no free variables. Therefore, $rank(A) = n$.

Answer 1

$Ax = b$ has a solution for every $b$ is equivalent to saying that $A$ is surjective (where I'm viewing $A$ as a linear function $A : \mathbb R^m \to \mathbb R^n, x\mapsto Ax$), so $\text{rank} (A) = \dim (\text{Im}(A)) = n$.

What exactly do you mean by $Ax=b$ has a unique solution. Do you mean that there is only one $b$ where $Ax=b$ has a solution or that for every $b$ there is exactly one solution to $Ax=b$?