Not sure what “links-style” is but the “links” I like are Augusta and Pebble Beach and even Stanford where trees and natural surroundings make the courses attractive if not spectacular.
To me barren “links” look like the old Palo Alto Municipal Golf Course in front of the municipal airport.
Instead of playing trees and water u play wide open in the wind and tall rough.
Links-style describes those treeless courses you see in Britain. I believe it is the oldest style and is based on the natural terrain found along the coastlines. They were developed at a time without golf architects and earth-moving equipment, so they reflect a kind of minimalist design. They have few trees, tall grass rough, pot bunkers, and rolling fairways. I think every British Open is played at this type of course. They are very frustrating to play.