Do you correct people a lot? What's your motivation?

Yep. That accounts for quite a wedge of the corrections I see - people enforcing an imaginary rule.

Two in particular from the past week:

“it’s only shepherd’s pie if its lamb, otherwise it’s cottage pie” - simply untrue. The earliest shepherds pie recipes used any meat that was available, and besides, shepherds are not people who eat sheep.

“You must not start a sentence with a conjunction!”
And why not, exactly? Or is there actually a rule?