Robert Price defines a links as land that is immediately inland from the sea, underlain with sand and gravel, and comprised of hollows, ridges, and mounds. He defines it briefly in his 2002 Feature Interview for GCA. I am guessing his book has a more extensive definition of the term. From that definition, I think it would be impossible to have a non-links course on links land.
In my understanding, a course is deemed to be a links by the type of land it sits on. It has nothing to do with the design of the golf course. Many golfers in the US believe any sparsely-treed golf course is a links course. These people are way off base. A links course must sit beside the sea on sand and gravel soil. The land must be low-lying but undulating. Any course that does not meet these requirements is not a links course.