Only 1 of those is a true self opener. That’d be the Beesley action. Beesley designed two SO actions. The Purdey and Lancaster. You’re probably referring to the Purdey type.
The H&H is more of an assisted opener type mechanism, the heavier the barrels on the H&H system, the better the system works. You’ll notice a big difference on how much opening action there is on a H&H when the gun is unfired vs fired. You won’t notice any difference on a Purdey, except when you close a fired action.
The Purdey action is more complex than the H&H. The ejector mechanism on the Purdey action (WEM) is more complex than the common southgate used on the H&H. Many more gunsmiths are familiar with the H&H action than the Purdey style. Purdey style requiring some specialized tools & skills to work on the gun.
Brief, but I hope this helps.
