A common misconception. The Higgs mechanism gives the fundamental particles (such as quarks and electrons) their masses, but most of the mass we see comprises protons and neutrons, whose mass arises primarily from quantum chromodynamics. (Dark matter, which dominates over the "baryonic" matter I mentioned, may or may not get its mass from the Higgs mechanism.)
I'm not a physicist, but you do sound like a pretty reasonable scientist, to me, to use this phrase.