Carbon taxes have a substantial base of support on the left (possibly more than cap and trade, whose base of support is more centrist.)
Now, carbon-tax supporters are likely to support cap-and-trade as better than nothing, but that's different than preferring it and opposing carbon taxes.
Lots of left-wing ideologies favour using market mechanisms this way, and I wish more would - you can want to use market-mechanisms this way whether or not you're for free-market competition in terms of valuation and trade of company shares.
In reality, the two "choices" imposed upon us are Coke and Pepsi, two noxious compounds that require extended chemical analysis to differentiate. They both want more war, all the time, in every case, for whatever reason. The real reason, of course, is so that they may skim some of those sweet armaments manufacturing profits. No it's not a conspiracy. Important influencers know which side of their toast is buttered, but they are a small minority and they don't need to coordinate in secret. Everyone who doesn't revolve from bureaucratic post to public office to lobbyist to executive board to think-tank sinecure, is a mere useful idiot. If 10-20% of American voters ever learn about horseshoe theory, they'll be screwed. Therefore the media mentions that about as often as they mention e.g. Libya or Yemen.
So, it's little surprise that we Americans have no clue about leftist thought.
Nothing was conflated.
I also don't trust government bureaucrats to figure this all out, either.
A carbon tax basically would just reward good ideas in the market, and punish the bad ones.
However, almost no one supports it. You get some big figures like Musk and the economist, Mankiw; but both the left and the right of mainstream politics don't want it. The right doesn't want new taxes nor to particularly combat global warming; the left is ideologically opposed to market-based solutions and instead supports big government/power-grab solutions--specifically cap and trade.
TL;DR consider supporting a carbon tax if you like these ideas and want to combat global warming