Mark as ever you are correct.
Another admirer of Woking was Guy Campbell the architect of Pulborough "Woking possesses more good strategic holes than any other inland course" and "I never cease admiring the genius which created these holes, especially when on remembers that they were made long before the southern golfer had been educated up to this kind of thing".
John Laing Low was captain 1901-1902, Stuart Patton (“The Mussolini of Woking" Darwin) was Captain in 1902-3. I would guess the changes started during their tenure. The fourth is commonly referred to as the first strategic inland hole. The recent book “The Centurions of Golf” described their work as improvements to “Dunn’s somewhat simple layout. Redundant bunkers were removed and replaced by grassy hollow, undulations and subtle slopes were built into Dunn’s flat, square greens, and tees were repositioned.” Finegan says it was cross bunkers that were eliminated.
As to the influence of the club it had many famous members. Low was influential with the R&A, particularly on the Rules Committee (it was only in 1897 that the R&A’s version of the Rules were universally accepted.); Darwin became a member in 1897. Balfour was captain 1904-5.
Low wrote in Concerning Golf (2nd ed. 1906) about a club with which he was connected recently adding a bunker at the 17th that aroused bemused comments from members.
We have to remember it was the first course built on the free draining Heathland close to London; as such it would have been watched and much admired. Its timing couldn’t have been bettered when it was founded in 1892 it was at the cusp of an explosion in interest in Golf. Being close to a railway line meant it could easily attract a wealthy and influential clientele from London and right from the start it attracted golfers of real historical interest to join. I can find no record of it hosting tournaments and this may have led to it becoming less well known than say Worplesdon, Walton Heath or Sunningdale. I would love to see a copy of the History.