Yes, it's true that all else being equal, a beam with 1/4 the width would be 16 times brighter.
However, in reality 'all else being equal' is a pretty hard standard to live up to. The profile of brightness across the beam width can vary significantly, and this will have significant impact on the ratings. A profile that's flatter across the center of its width will have lower peak intensity, while one that is more peaky at the center will tend to have higher peak intensity. And such characteristics are not easy to design into optics. You tend to get what you get. To get something else tends to be very expensive.
So a subjective comparison might correlate well to the theoretical numbers, while a measured, objective comparison might not correlate as well. Or vice versa.
With that said, I'm seeing something deeply wrong about the numbers you posted. It appears that you said the lights claim to produce 18,000 lm from 18W. According to the laws of physics, this is not only impossible, but it's quite far from what is possible.
18,000 lm from 18W is 1000 lm/W. Physics says the maximum possible is somewhere around 340 lm/W for white light. IIRC the max is for a specific green color, something like 560 lm/W.
Anyone claiming to produce 18,000 lm from 18W is quite frankly lying to you. Which begs the question 'what else are they lying about?'. Welcome to the world of marketing.
