I drink a lot of caffeine, will this register calories since my heart rate is elevated during this time?

You'd have to be drinking very large amounts of coffee to raise your heart rate enough to cause the Apple Watch to register it as exercise.

Is the "pacer app" a third party app? If so, it's probably as simple as a difference in how it calculates caloric burn. Keep in mind that, at best, devices like your watch, third party apps, the equipment at the gym are making estimates.

See this article:

Get the most accurate measurements using your Apple Watch - Apple Support

You can recalibrate your watch. If that doesn't improve the accuracy of the readings, reset your watch by pressing in the crown and side button until you see the Apple logo, then release both.