The residual for a candidate $x$ is $r=b-Ax$. Its Euclidean error is $\|r\|_2$; the sum of squared errors is $\|r\|_2^2$. Both objectives have the same minimizers. The minimum error is the distance from $b$ to $\operatorname{Col}A$, not necessarily the distance between fitted coefficients and unknown true coefficients.
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