From what I gather from reading different reviews of golf courses, a links-style course has:
• rolling, sparse fairways
• preferably along the coast
• tall, weed-like grasses and pastures
• large greens
• pot bunkers (whatever that means)
• ill-defined fairways
Historically, it sounds like a links-style course was really just some lousy land near a coast that was converted to a course. The play is atrocious but I guess that gave bragging rights. I think most courses just use the word because it sounds cool or they have one characteristic, sparce, rolling fairways.
I participated in the planning for a golf community around a links-style course and, IIRC, the difference is that instead of having large, continuous fairways from tee to green (think of all that green grass at the Masters), a links course has much smaller areas of fairways broken up by unprepared ground. You play from “landing area” to “landing area” to green. Add the deep pot bunkers (bunkers that are so deep you can’t see out of them) and you’ve got a long, frustrating day ahead of you.