Was this damage to my ceiling caused by heat from a water heater vent?
This is common in water heater vents.  That goes up and eventually goes through the roof.  If installed properly the roof vent should last a long time and not leak, but... stuff happens.  They leak from time to time, and the leaking water will simply run down the pipe until it hits that drywall you see.

The yellowing isn't from being burned, is just dirty water seeping through the drywall which then loosens the texture on the ceiling.

So, you have a roof leak, not a faulty water heater.