free standing pergola footing
I would expect embedding the post in concrete to give the best resistance to upward lift and it for sure provides better resistance to lateral loads. However, it has the clear disadvantage of subjecting the wood to long term contact with moisture. If you're using cedar or below-grade rated PT lumber, you'd think that's not a concern. Well, you could spend a day googling that debate and you'll probably end up where you started--it depends who you ask. I can tell you I have plenty of experience with PT wood embedded in the ground and rotting after anywhere from 6 to 15 years. So, based on my experience, I never let a post touch soil or concrete unless I absolutely need to. E.g., with fence posts, you don't have an option.
For a pergola, I would keep the posts above ground and use any of various common techniques to resist lateral loads (you mentioned one).