I personally find that it's not so much that we are necessarily capturing the "personality" of the pet, but rather the mannerisms and facial expressions that are easily attributable to the pet and recognizable to the pet parent. I love it when someone says something to the affect of "My dog makes that funny expression with her ears perked up whenever we're about to go on a walk - I love that you got a photo of that!"

I always strive to get as many natural expressions and actions/postures shots as possible. I personally will not know which exact expression/mannerism will bring out that gut reaction from the client, but hopefully with enough variety, I'll hit the mark.