[swarmee]

No, you've got it wrong. People who collect the PSA set registry want their cards graded by PSA and are usually willing to have a slightly lower grade in a PSA slab. So by crossing over the card, if it meets the min grade requested, PSA will take it out of the BGS or SGC slab and put it in their own with the new PSA grade. Makes their collection more uniform if they are working on building a set with one company. Sometimes you can make a profit crossing, some time you'll take a loss.
Getting a card graded over a 5 from any grading service should ensure the card is crease-free, so knowing that will help you get a similar grade from PSA. Each third party grading company has slightly different things that will set them off: PSA rarely grades cards nowadays that is factory cut short. BGS will usually grade sheet cut cards (modern laser cuts of old complete sheets). SGC will hammer cards with writing on them, while PSA will give them a MK qualifier.