In the Higgs mechanism, the W and Z acquire mass by pairing up with degrees of freedom from a Higgs field. (Fermions get their mass differently; a purely left-handed massless fermion and a purely right-handed massless fermion pair up into a single massive fermion via their interactions with the Higgs field, but the massive fermion does not incorporate degrees of freedom from the Higgs field into itself, the way that the massive gauge bosons do.)
A Higgs boson comes from the unused degrees of freedom of a Higgs field. In the long run, the new particle will be called a Higgs boson if, and only if, it continues to look like a degree of freedom left over from a Higgsing. So it can still be a Higgs without being a standard model Higgs; for example, it might be one of several Higgses, or its couplings to the other particles might be different from the standard model values. The fermions could get their masses in some other way entirely (e.g. by being composite), but this particle would still be a Higgs so long as it’s the residue from the acquisition of masses by the W and Z. That’s the aspect which has to be false, if it is not a Higgs.
In the Higgs mechanism, the W and Z acquire mass by pairing up with degrees of freedom from a Higgs field. (Fermions get their mass differently; a purely left-handed massless fermion and a purely right-handed massless fermion pair up into a single massive fermion via their interactions with the Higgs field, but the massive fermion does not incorporate degrees of freedom from the Higgs field into itself, the way that the massive gauge bosons do.)
A Higgs boson comes from the unused degrees of freedom of a Higgs field. In the long run, the new particle will be called a Higgs boson if, and only if, it continues to look like a degree of freedom left over from a Higgsing. So it can still be a Higgs without being a standard model Higgs; for example, it might be one of several Higgses, or its couplings to the other particles might be different from the standard model values. The fermions could get their masses in some other way entirely (e.g. by being composite), but this particle would still be a Higgs so long as it’s the residue from the acquisition of masses by the W and Z. That’s the aspect which has to be false, if it is not a Higgs.