Here’s the theory…
The focus really sets both the height and the width of the waist of the hourglass shape.
A shorter focus gives a narrower waist but a shorter depth of focus, and is relatively more useful for engraving and worse for cutting. A longer focus gives a wider waist but a longer depth of focus, and is relatively more useful for cutting and worse for engraving.
The ramp test is the measure of focus, because the only thing that matters about focus is what it does to the material. So in the ramp test you care first about optimizing the beam for engraving or cutting. If you are engraving, you want to know the height at which the line is thinnest, and it’s OK if the depth of focus is shallow as long as you are engraving something flat. If you are cutting, you want the depth of focus to support cutting through the entire thickness of the material.
The difference between theory and practice
The reason you don’t see much discussion of this for LED lasers is that the entire thickness of material they can cut through is pretty low, even with air assist. So in practice I haven’t heard of people refocusing their LED lasers for cutting, because most of the time people who want to cut are not using LED lasers.
So, do ramp tests for focus, and adjust the focus for the workplane you want.
(I understand that not all LED lasers even have adjustable focus; for those you just use the ramp test to decide where the workplane needs to be.)
I’m not talking you out of high quality laser safety glasses! The blue lasers are super dangerous to your eyes because they will fry your retina without you noticing. (My understanding is that CO2 lasers at least blast your cornea first…) I’m only suggesting that trying to focus by looking at the beam isn’t a good approach.