Active calories in recorded run (not including resting calories) are higher than active calories for the day.

Is the all-day heart-rate graph (not the one of the activity!) complete for the concerned day, or is there a gap during the activity in it?

Yes, no gaps. Heart rate is continuous including the activity time. The same thing happened today. A ten mile run said 568 active calories but in the calorie count it only says 510 active calories. 

First of all, I'd try soft-resetting the watch (hold the Light button, till the watch shuts down, then restart. No data will be lost). If it does not help, the master reset to factory defaults hopefully can (all settings will be lost, but data remains preserved in Garmin Connect). If still no change, contact the Support. You can also check the forum dedicated to you watch model, to see whether other owners of  the same model have the same problem.

Eh - I forgot to tell - there is a possibility that some parts of the data land in a phantom account. That's a rather common problem. You should also try following the instruction given in the hint #4 of the  Troubleshooting guide for (some) data missing after syncing. It could help.

Awesome. Thanks so much for your help! I'll try both.

Just edited my post, Have a look at the last tip definitely too