How can an electric induction cooktop be safely installed into a diy natural wood countertop?
My GE induction cooktop has almost the same warning but it only specifies wall coverings and cabinets (not the countertop). The walls and cabinets can get direct radiant heat from the cookware but the countertop is unlikely to get quite so hot simply because of the angle of incidence (which reduces the intensity of the radiance). The cooktop itself shouldn't generate such high temperatures (e.g., from it's internal cooling fans).

Metal flashing can protect against flash fire or embers (and is specified if there are wood cabinets 24-30" directly above the cooktop. But this won't change the impact of continuous radiant heat. So lining the cutout hole with flashing would not make a difference. (Surrounding with a wide metal "U" channel would make a difference but you'll see I don't think its necessary.)

The jury is still out on whether continuous exposure to temperatures above 170F can lead to combustion of wood due to self-heating (pdf). This would be relevant for wood surfaces near a pellet stove or steam pipes running through a floor opening but I have difficulty believing you will generate these high temperatures for prolonged periods with a residential induction cooktop. It's simply not designed to operate continuously for long periods. (If this is a DIY restaurant than you'd need to rethink things.)

You should observe charring (over time) near the cooktop prior to any ignition due to self-heating. In the very unlikely event this happens then I would consider widening the cooktop opening and installing metal u channel or other material.

Overall, it seems the risk is low and you should not need to take any special precautions. Someone else might be able to comment on whether the residential building codes specify appropriate surfaces for a countertop (e.g., for flame retardance). Your countertop would need to meet any applicable building codes regardless of the cooktop.