Short answer: They discovered a particle which looks like a Higgs Boson.
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.
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.