"Carbon emissions" is a collection of gasses, one of which is CO2. It's pretty common to equate the damage different gases do to CO2 and refer to it as "Carbon", which then get's messy when people paraphrase and leave out some context. For example Methane does something like 80x more damage than CO2 by weight, so it said a ton of Methane is "80 tons of carbon"

And Science doesn't work by "Absolutely 100% proving" anything. Bayesian statistics which underpins almost all of modern science lets us take highly educated guesses with communicable confidence in that guess. So stop looking for "100% absolute proof" to be convinced, you won't get it.