The terms 'easy opening', 'assisted opening' and 'self opening' are used without having specific meaning.
They all relate to actions in which the barrels are pushed off the action upon operation of the top lever (or whatever lever is used).
The original 'self-opener' was the Purdey in 1880, designed by Beesley. this has a strong self-opening feature powered by both mainsprings.
The 1922 Holland & Holland patent has coil springs in a tube under the forend to assist the opening of the gun.
The Lancaster self-opener as a sidelock and a boxlock are variations on the Beesley patent.
Each and every one of the assisted and self-opening mechanisms vary and some are stronger and more reliable than others. Weigh up each one on its merits and see if the operation suits you.