Without getting into truths of earlier discussions, let me try to restate in another way.
The force for the spring is equal to spring constant x compression of spring (in this case). When the valve is closed, the compression of the spring (reduction in original spring length) results in a force exactly equal to the set pressure for the spring at which the valve will simmer as operating pressure increases - and then pop open due the large disk area (huddling chamber). This assumes no superimposed back pressure; which if present increases the opening pressure proportionally because it operates on the back side of the huddling chanber.
Now, since the spring force increases with further compression of the spring as it begins to lift, the huddling chamber must be present to assist in opening the valve; and the 10% overpressure is specified to obtain full opening of the valve and to guarantee accurate calculation of the required orifice size (ie - basis for equations used).
So much for balancing forces, valve stays open so long as relieving pressure on huddling chamber results in a force greater than spring force (Lets leave backpressure out of argument until a bit later). Note that the valve will be less than full lift as relieving pressure decreases until it closes at the reseating pressure the valve is set at. Also note that when the valve is at full lift, there is no further change in spring force since there is no further compression of the spring - there is only an increase in contact force of piston on valve body. If upstream pressure increases, contact pressure increases and flow increases.
Now lets talk about backpressure or should we say variable backpressure since all have agreed constant backpressure translates directly and is additive to the spring set pressure.
Next, for sonic flow occuring at the orifice. The combined total backpressure (superimposed and built-up) will determine if the capacity of the valve is affected and the vendor publishes charts for their valves which desribe this relationship (Kb). Its not as simple as if pressure drops to less than 1/2 the inlet pressure, further pressure reduction will not increase flow. While total backpressure reduction will not increase flow, increases in backpressure will reduce flow to less than certified capacity (specified at 10% normally). But allowable backpressure can be quite a bit higher than 10% without a derate in capacity and is related to how much is variable versus constant and the type of relief valve (balanced, conventional or pilot-operated). In general, however, we usually try to design for 10% builtup backpressure max with a conventional valve, although this can be exceeded, before we go to a balanced bellows valve. At the other extreme, backpressure might be as high as 30% for a bellows valve; or even 60% constant backpressure for a conventional valve relieving gas or vapor without a capacity reduction being required by the vendor. Use the vendors CHARTS! THey were developed to account for backpressure on a orifice at sonic conditions. Remember even though the capacity may be decreased, the orifice is STILL AT SONIC CONDITIONS. Why make it any more complicated? Use the vendors CHARTS!
The more you learn, the less you are certain of.