Clarifying the actual definition of elasticity. Is steel really more elastic than rubber?

Both the OP and John Rennie have well illustrated the imperfections in the usage of the word "elastic" in physics and how the word can create confusion between "stiffness" and a material's ability to brook strain.

But an important point to be made is that the one important field where one hears the vague statement that "steel is more elastic than rubber" is in the context of Newtonian collision problems. So what's meant here is that steel objects typically undergo more elastic, i.e. kinetic energy conserving, collisions than rubber ones.

Newtonian collision problems come up very early and prominently in an undergraduate physics course, so this may well create the (probably mistaken) impression that physicists tend to mean stiffness rather than ability to brook strain by the word "elastic". Indeed, the fields wherein physicists, as opposed to specialist material scientists, mostly use the word "elastic" are those where the word refers to collisions and interactions, and in these contexts the word means "conserving of total kinetic energy of all colliding bodies" or "not energy converting". Elastic optical interactions such as Rayleigh scattering or Fresnel reflexions are those where the incident and scattered light have the same wavelength, thus photon energy, and no energy is dissipated in or transferred to the scatterer. Likewise with all particle physics specialities.

An interesting comment from user Jasper:

In other words, rubber's stress-strain curve has more hysteresis (as a fraction of the maximum strain energy in the loop) when the strain goes from negative to positive and back.

Intuitively, it's probably part of the cause, maybe the main cause in some materials but there are rubbers where other mechanisms account for the loss, according to some cursory research I've been doing into rubbers in recent weeks for bearings in an adaptive optics system I've been working on. I'm certainly no expert, but common models used are all linear differential equation models wherein the loss comes from damping terms. Look up the Kelvin Voigt Model and Maxwell-Wiechert Model and Standardized Linear Model. Synthetic rubber manufacturers often specify the loss properties of their wares by loss tangents and complex-valued Young's modulusses (which show a phase delay for sinusoidal force excitation). Mechanisms other than hysteresis that can give rise to loss tangents are viscous drag between neighboring molecules; this can be simply linear damping of the form −μx˙$-\mu \dot{x}$ where x$x$ measures the strain and μ$\mu$ a viscous drag term. To be clear: by "hysteresis" I mean a nonlinear, instantaneous two-valuedness of a strain-stress response curve where which of the two function branches is traversed is set by the direction of the variation. Each cycle around a Bvs.H$Bvs.H$ loop in a ferromagnetic material or around a σvs.ϵ$\sigma vs.ϵ$ loop in a deformable material transfers energy proportional to the loop area to the material. This is different from viscous drag.

My interpretation of the statement "steel is more elastic than rubber" is very different from yours.

I would say that it means rubber is viscoelastic and that there is a time dependence to the stress-strain relationship; it flows when you shear it. Steel will be very nearly perfectly elastic until reaching yield.

Understood this way, we can say that for a given stress OR strain applied, rubber will never be perfectly elastic. This is, by the way, basically equivalent to saying that no energy is lost in elastic collisions, as that energy is going into rearranging long chains of hydrocarbons in rubber instead of just vibrating an iron-carbon lattice and slightly heating it up.