One way to characterise the difference between these two notions of equity is to introduce the notion of "justice." Thus, the term equity is used in the traditional sense of who should bear what share (and in what manner) of reducing their emissions of GHG, while the notion of justice (or more correctly "injustice") is what characterises the argument of the "victims" of climate change against all major emitters.
Of course, in large developing countries, like China and India, where there are large populations of poor and vulnerable citizens living alongside smaller numbers of rich citizens (many of whose per capita emissions are equal or bigger than those of rich countries) there is a debate to be had as to whose interests their governments are representing when they refuse to take binding commitments to reduce emissions of greenhouse gases from economic sectors that largely benefit the rich and urban middle classes, but which will contribute significantly to the adverse on the rural poor of their own country. This is a debate that citizens of China, India, Brazil and South Africa need to have.