They are talking about the standard model Higgs, which is the result of a specific way to break electroweak symmetry. A consequence is, that there are quite well understood predictions from this how the cross sections and branching ratios should look like. And on the current level of statistical significance it looks like a standard model Higgs.
On the other hand, there are so called effective field theories, that is you can start from a complicated theory and derive a simpler theory from it, which behaves the same in some aspects (for example at low energies).
So the more exact answer is probably that now a theory has to contain a Higgs boson in the appropriate limit.
Since the funding dried up for the Tevatron and the first hints for the discovery of the Higgs boson come from LHC we can conclude that Hawking won the bet and Kane will be the one paying up.
Clarification:
http://public.web.cern.ch/public/en/lhc/LHCExperiments-en.ht...
"In this Letter, we report on the combination of Higgs boson searches carried out in proton-proton collisions at Sqrt[s] = 7 TeV using the Compact Muon Solenoid (CMS) detector at the LHC. ... Combined results are reported from searches for the SM Higgs boson in proton-proton collisions at Sqrt[s] = 7TeV in five Higgs boson decay modes: gg, bb, tt, WW, and ZZ."
Since this is apparently "only seen" through the Monte Carlo analysis I suppose they haven't seen one specific particle decay with it, I suppose?!
Atlas comes in at ~126.5 GeV with 5.0σ. That would be a confirmed discovery!
Interesting that Atlas' mass is outside CMS's confidence range, though Atlas didn't have a range on theirs.