What is stiffness in a material

The modulus of elasticity (E) is not known as stiffness. Stiffness for an axially loaded compression or tension member, as you noted (which you defined as s, but which is typically noted as k as in Hooke's law F=kx), is AE/L. Beam bending stiffness is a function of E,I,L and boundary conditions . E by itself is a property of the material only, which is a measure of how much the material will strain under stress.

For example, steel has a modulus E which is 3 times greater than aluminum. But a short piece of aluminum might have a greater stiffness than a long piece of steel.

 

Stiffness is usually defined by flexural modulus.

 

The confusion is in the term 'stiffness' and how it is defined.

Based on properties of the material alone, steel is, 'materially' speaking , stiffer than aluminum, since it has a higher E modulus, and will strain less than aluminum under a given stress.

But based on the properties of the structural member, including E, A, L, and I, then, 'structurally' speaking, a short/stubby piece of aluminum will be stiffer than a long/slender piece of steel, since it will deform less under a given load. This definition is consistent with Hooke's law definition of stiffness for axially loaded members.

Actually, the conclusion reached by the OP is excellent.

Regarding the question of the units for Elasticity,which is stress/strain, it must have the same units as stress since strain is a dimensionless quantity. I think of elasticity as the stress in a member that would exist at 100 percent strain, that is, the stress in the member that would exist when it was elongated by a force causing it to stretch twice it's original length ( assuming a perfectly elastic material of perhaps unimaginable strength).