There's no evidence that one way of removing a bee stinger is better than another. I could only find one published study that looked at the issue. It compared the difference in reactions between bee stingers that were removed by pulling and bee stingers that were scraped off. It's not a great study because it was a very small number of stings being compared, but I think it made the point.
In that study, the only thing that mattered was time. Pinching the stinger and pulling it out didn't cause more of a reaction than scraping it off with a credit card. What did cause a bigger reaction was leaving the stinger in the skin longer.
The moral of the story is to do what animals do: Get the stinger out! When animals get stung, they don't reach into their wallets for their library card to scrape out the stinger. They bite it off or rub up against a tree or scratch with their paws. Animals instinctively know that if it hurts it needs to go, the sooner the better.
Visscher PK, Vetter RS, Camazine S. "Removing bee stings." Lancet. 1996 Aug 3;348(9023):301-2.