If you have POE going through there that could cause that type of damage I would suspect.

Too much P over the E

More like too much "I".
pasted_image (resized).png

1) Is the connection point exposed to weather?
2) Were the ends of the wire already terminated, or did you (or someone else) terminate them? Either way, there's a possibility that the wires in the ethernet cable weren't terminated correctly. Here's a cheap ethernet cable tester: https://www.amazon.com/AOWIZ-Network-Cable-Tester-Ethernet/dp/B09F68WGLK That should be able to tell you if there's a short or any crossed wire pairs.
3) What device is providing PoE power? The recording unit? A separate switch?

Probably, although it might be possible to swap the two positive wires carrying voltage.

I wouldn't until you test the cable. It might be terminated correctly, but there might also be a rodent problem in the middle, like you said.

So, as a precaution to avoid frying other equipment, I'd test first, then if that checks out, go ahead and swap.

The recorders POE output could be the issue and/or the cable. Being that the cable is not as long as some of the other cameras that still work it's likely the power source (recorder). Replace any bad ends and maybe try a separate POE injector

https://www.amazon.com/Injector-Compliant-Compatible-TL-POE150S-TPE-113GI/dp/B09BFBM6PQ/ref=asc_df_B09BFBM6PQ/

Looks to me like you're not getting a good connection on the power pins in the camera pigtail, leading to arcing and burning. Did you re-terminate the cable after the first incident? If not I bet the gold plated connectors in the cable are shot. If so, maybe try re-terminating with a different brand RJ-45 plug? Make sure the cable is good and snug when clicked in.

I've seen SATA HDD power connectors burn up during normal use just because they didn't have a good grab on the edge connector... they overheat and burn, just like the old J122 connectors for WPC GI.

Maybe the DVR is overloaded and the cameras are drawing more power than it can handle?

IDK. Seems like either cheap china connectors or some kind of design issue with the DVR setup.
I thought all RJ45s had gold plating on the crimp connector. It's really "odd" to me that they choose to use tin plated "Wipers" in the female end. Maybe that's part of the issue.

What did the camera's tech support say when you contacted them about burnt connectors?
Maybe it's a known problem they don't want to admit?

You could absolutely try another port on the recorder. The POE injector will be a more robust power supply than your recorder. If it's easy to do I would also completely replace the Cat5 cable and if available use Cat6 (it's slightly thicker copper and wound tighter for faster data transmission). I've had Cat cables go bad on projects. Although it's not common I have definitely had it happen to me especially on outdoor cables. Be sure to use a direct burial cable. A standard cable's jacket will deteriorate over time and water can get in there.