I've had installers provide certification documents for new cable installs, which they supposedly tested and had bad runs which required replacement of cable. Maybe someone tested a cable twice, or BS'd, who knows. When they certified the cable, did they certify it for data, for POE, or both?
Again, I'd suggest re-terminating the cable ends and/or putting a POE tester on it which can provide more insight into what's on the wire. Your camera obviously isn't getting the right power via that cable. Logic to me would say if switching out the cable with an exact same cable (length, type, manufacturer, etc.) works, then it's most likely a bad cable.
Let's not get hung up on the vernacular, TIA/EIA-568B (or C now, supposedly), is the standard I'm aware of and use for cable installs. That standard defines the pin to pair assignments. Add to that POE uses positive and negative current on separate pairs of cable, so I use the term "pair."
Strictly speaking, though not good practice, as long as the cable is pinned out in a method which is compatible with the switch, POE injector, and camera; that's all that matters. A POE tester should tell you that, and if there's anything going on with the cable (e.g. damage) which prevents it from carrying the POE.