This paper refines earlier results from [\textit{K. F. Ng, X. Y. Zheng}, SIAM J. Optim. 12, No. 1, 1--17 (2001; Zbl 1040.90041)] on error bounds for lower semicontinuous functions \(f:X\rightarrow {\mathbb R}\) defined on a metric space \(X.\) In particular, the authors consider error bounds with exponent \(\beta >0,\) in which the distance from \(x\in X\) to the set \(S:=\left\{ x\in X:f\left( x\right) \leq 0\right\} \) is compared with \(\left[ f\left( x\right) _{+}\right] ^{\beta },\) where \(f\left( x\right) _{+}=\max \left\{ f\left( x\right) ,0\right\} ,\) instead of with \(f\left( x\right) _{+}\) as in standard error bounds. They also extend another characterization of error bounds for continuous convex functions on reflexive Banach spaces in terms of subdifferentials obtained in the above mentioned paper to the lower semicontinuous case in a nonreflexive setting and give new characterizations, some of them in terms of lower Dini derivatives.