Streaming codes (SCs) are packet-level codes that recover erased packets within a strict decoding-delay deadline. Streaming codes for various packet erasure channel models such as sliding-window (SW) channel models that admit random or burst erasures in any SW of a fixed length have been studied in the literature, and the optimal rate as well as rate-optimal code constructions of SCs over such channel models are known. In this paper, we study error-correcting streaming codes ($\text{SC}_{\text{ERR}}$s), i.e., packet-level codes which recover erroneous packets within a delay constraint. We study $\text{SC}_{\text{ERR}}$s for two classes of SW channel models, one that admits random packet errors, and another that admits multiple bursts of packet errors, in any SW of a fixed length. For the case of random packet errors, we establish the equivalence of an $\text{SC}_{\text{ERR}}$ and a corresponding SC that recovers from random packet erasures, thus determining the optimal rate of an $\text{SC}_{\text{ERR}}$ for this setting, and providing a rate-optimal code construction for all parameters. We then focus on SCs that recover from multiple erasure bursts and derive a rate-upper-bound for such SCs. We show the necessity of a divisibility constraint for the existence of an SC constructed by the popular diagonal embedding technique, that achieves this rate-bound under a stringent delay requirement. We then show that a construction known in the literature achieves this rate-bound when the divisibility constraint is met. We further show the equivalence of the SCs considered and $\text{SC}_{\text{ERR}}$s for the setting of multiple error bursts, under a stringent delay requirement.