In that case it seems like "HQI" is the same as xerox, kleenex or Coke; an actual HQI bulb is an Osram manufactured light, vs. say an Ushio or other SE or DE bulb that would simply be called a metal halide lamp.
What I find interesting is that osram.com shows the definition of a metal halide lamp, but doesn't show anything about an "HQI" lamp:
Metal halide lamps
Metal halide lamps are high pressure mercury lamps with added metal iodides or iodides of the rare earths (dysprosium (Dy), holmium (Ho) and thulium (Tm)) and complex compounds of caesium (Cs) and tin (Sn). They decompose in the core of the discharge arc and the metals can be excited to emit light with an intensity and spectral distribution that depends on the vapour pressure of the metal halides. The luminous efficacy and the colour rendering properties of the mercury discharge are significantly improved as the gaps in the mercury spectrum are filled by the spectral contribution of the other metals.