Bardic performance is listed under the bard class description as a single class ability. (On the PRD, each class ability begins with the emboldened name followed by a colon.) The individual performances are ways in which you can utilize that class ability based upon your class level.
Based on that, you gain access to new performances as you gain levels in both of the prestige classes you mentioned.
(Note that in the case of Pathfinder chronicler, your effective bard level contribution from that prestige class is two less than the actual number of levels in said class--e.g., a bard 8/Pathfinder chronicler 8 would have bardic performance at an effective bard level of 14. Dawnflower anchorite has no such subtraction--you select the credence, you gains the boost.)