Not sure exactly what you mean, since stress is stress, effectively. Even the Wikipedia article on flexural strength puts it thusly: "the flexural strength would be the same as the tensile strength if the material were homogeneous" i.e. if it's different, it's for specific reasons related to the particular material, not as a general rule. I've never heard of more than one Young's Modulus (E). (There is a Shear Modulus, G, but that's a different question.)
You say "I know the modulus from flexural strength is measured from load/extension rather than stress/strain of compressive strength", but there's a reason why Young's Modulus is defined in terms of stress/strain - because those are "normalised" and consistent w.r.t. the material.
stress = load / area, and it's stress that causes it to fail, not load. The load can be increased, and the cross-sectional area increased, resulting in the same stress.
Likewise: if you apply a particular stress to an elastic material you get an extension that depends on the length, for any particular section. So we divide the extension by the length to give us a strain figure. So we find that, for any given elastic material, regardless of length or cross-section area
, stress and strain are proportional. The proportional constant is what we call the Young's Modulus (E).
It doesn't really matter whether that stress was created by axial force or by bending - it's still stress, you just calculate it in a different way, and you can have both axial and bending at the same time, so the stress adds up. The formula I know for maximum stress under both is:
where the first term is the axial stress (normal force over area), and the second term is stress generated at the outer edges by bending. (M = bending moment, Z = section modulus = I/y).