Springiness and flexibility are two different descriptions of the same event.
Springiness is the ability if something to bend and return without distortion. Flexibility is the amount of bending needed before permanent distortion or breaking.
The flexibility of steel is a fixed factor, determined by Young's Modulus of Elasticity. How much force will bend a certain thickness of steel.
What you can can control is how easily the bending leaves a permanent bend ( plastic distortion) or breaks ( failure). The hardness of the steel , and the method of tempering will control these things. The type of structure in the steel ( austenite, martensite, pearlite, bainite) ,or the mix of them, will be how we do the controlling.
The main way for knife/sword people to increase or decrease flex is to increase or decrease thickness. The other geometry features, like distal taper and bevels, confine the flex to certain areas.
Kevin regularly demonstrates how to do a 90° bend test on a blade and have it return to dead straight. He even does it twenty times in a row, and the blade is dead straight.....
The blade is a double edged razor blade, and he holds it in his fingertips, pinching it to bend it 90 degrees.
I have a 16th century sword blade I take to knife events. I hold it over my head by the ends, and bend it in a "U". It springs back.....Not because ancient smiths had superior powers in forging, but because they made thin sword blades.
Sunshaddow (Page), makes live fencing blades and theatrical props, and will probably chime in soon.
Movie swords are made different. The theatrical blade should bend, but taking a set is preferred over breaking. They are softer and usually thinner.