First world is actually a term that goes back to the Cold War days and doesn't make much sense to use now. All countries ought to be held responsible for their carbon emissions because it's a societal cost that no one has an incentive to take into account. This includes wealthy countries.
If you dump pollution into the environment, you should be the one to clean it up. That's just common, logical sense.
From a practical standpoint, we're the only ones able to do anything about it anyway. Global warming has a high probability of causing an extinction event very soon if we stand by and leave the carbon in the atmosphere that's there already. It's going to take ridiculous technological development, and huge government funded programs, to put even a dent in the carbon that's already there.
If first world nations do not own up to the fact that they have emitted the most carbon throughout history, then it sets the precedent that no country should be held responsible for carbon emissions. How can the developing world be expected to cut back carbon emissions when the first world does not? There must be some level of responsibility and acceptance that there is a problem with carbon emissions.
Countries should be judged by their per capita carbon emissions, as that is one reflection of how they treat the world environment. Emissions affect everyone, and places like the US and Europe and China (as it grows) negatively affect countries like Maldives and Mauritius, for example, with low coastlines that are endangered because of rising sea levels.