While I don’t doubt that it helps with branding in the modern day, the name natural gas entered the public vocabulary centuries ago to differentiate it from synthetic gas/coal gas/town gas, not as a think tank branding exercise.
Created as a byproduct of the coking process, the aforementioned syngas was used primarily for its bright white light, and indeed in the US much of the network was built out after the invention of the lightbulb but before they got bright enough to be competitive.
Natural gas by contrast is produced by drilling into naturally occurring deposits of methane and other flammable gases as compared to being a combination of hydrogen and carbon monoxide produced by coal gasification. As electric lights got better and fewer factories needed to turn coal into coke, most cities decommissioned their syngas systems. A few decades later cites rebuilt gas distribution systems using natural gas to provide for a far more efficient form of heating, and people needed to easily tell the difference since syngas lighting and appliances arn’t practically useful with natural gas and so used the common name for it in much the same way most people don’t go to a restaurant and ask for a glass of high concentration dihydrogen monoxide, they just ask for water.