If you put,
VIT: Decreases Pet Revival Time. When a pet dies, it will take less time to revive itself.
Therefore, all you need to do is, add VIT! And, DO NOT TOUCH INT, because the more INT you have, the more time it will take for pet to revive.
So basically, if you want to Speed up the time it takes for the Blue Bar to load and your pet to revive itself, always train VIT!