The Links Golf Club (NLE since 1982) in Manhasset, NY was a separate (and newer) entity from The Links in NYC and NGLA in Southampton although most members of LGC (about 50 at its peak) were also members of one, or both, of the others.
I played LGC several times and it was an original Macdonald that had, basically, never been amended since it was built in 1918 - even by CBM, himself. I know some of the club's history although Tom Paul knows the most about it around these parts.
For what it's worth, the name of the NYC entity is, simply, "The Links" and not "The Links CLUB".