EDIT:
Ok, found some things. Google Image search for (no quotes) "clean natural gas" and set the range to 2000-2010. You'll find a variety of images both for and against this claim. Including things like "This Bus Is Running On Clean Natural Gas" (2009) [0] or "Clean. Affordable. Abundant. American. Natural gas is the answer. Tell Congress to put it to work." (apparently from 2010) [1]
[0] https://images.app.goo.gl/yfrQn8fYrB7zLqqh9
[1] https://images.app.goo.gl/4Aj7ybMt3wVWq49BA
Here's [2] an example from the 1980s calling it cleaner, source for me was [3].
[2] https://drive.google.com/file/d/1ympy04dETxXWfqkCn0zm9NmE1z1...
[3] https://www.theguardian.com/environment/2021/nov/18/the-forg...
But it was around the time of this bill, so it makes one wonder.