The particulars of "actual knowledge" are addressed in Viacom v. YouTube. The ruling from the appeals court states (emphasis mine):
Under § 512(c)(1)(A), knowledge or awareness alone does not disqualify the service provider; rather, the provider that gains knowledge or awareness of infringing activity retains safe-harbor protection if it “acts expeditiously to remove, or disable access to, the material.” 17 U.S.C. § 512(c)(1)(A)(iii). Thus, the nature of the removal obligation itself contemplates knowledge or awareness of specific infringing material, because expeditious removal is possible only if the service provider knows with particularity which items to remove...
Thus, actual knowledge must be knowledge that is specific enough to allow a service provider to take action against specific infringing content. "General" knowledge of infringement is not specific enough to be actionable, so it cannot qualify as "actual knowledge". (e.g., The general knowledge, "In the past, about 5% of the videos uploaded to our site were infringing, and we expect that trend continues into the present, so some of our videos right now are probably infringing," is not actionable to specific videos, so it does not quality as "actual knowledge".)
That ruling also clarifies the distinction between (i) and (ii), which are respectively termed the "actual knowledge" provision and "red flag" provision:
In other words, the actual knowledge provision turns on whether the provider actually or “subjectively” knew of specific infringement, while the red flag provision turns on whether the provider was subjectively aware of facts that would have made the specific infringement “objectively” obvious to a reasonable person.
Actual knowledge is different from a notification, which the service operator must respond to according to section (c)(1)(C) (whereas your quote is from (c)(1)(A)):
(C) upon notification of claimed infringement as described in paragraph (3), responds expeditiously to remove, or disable access to, the material that is claimed to be infringing or to be the subject of infringing activity.
Expeditious removal is required in either event: the operator has actual knowledge of infringement, per (c)(1)(A)(i), or the operator gets a notification, per (c)(1)(C). Actual knowledge and notifications are different, but they impose similar responsibilities for the operator.