These two seems like different manifestations of the underlying problem. Instead of thinking that when the laser turns on, it’s a sharp transition from “off” to “on”, there’s some function that represents the laser power output. Properly scaled, it starts at zero, goes to one, and takes some finite (and overly long) time to do it.
The difference between black and white paper is a difference in laser power delivered as heat to the paper. Black has lower albedo than white, which is almost a tautology as “albedo” basically translates as “whiteness”. In other words, a white paper reflects more of the incoming laser light, which means it heats up slower and crosses the charring threshold later. Black absorbs more energy, crossing the transition sooner.
Same for visible light. You see the laser beam because of Tyndall scattering, that is, there’s dust in the air and you’re seeing light reflected off of it. The eye has really wide dynamic range and isn’t finely calibrated for absolute intensity (relative intensity is a different matter). So you’ll see a beam at wide range of powers, including those not sufficient to burn.