Richard -
for me, no one element can define character. In fact, an almost certain way to ensure that a course has no character is to focus on/highlight one element in particular, thereby throwing off the sense/feel/look of unity and wholeness that is so appealing.
My home course is an early 70s public built on a farmer's field. Everything about it is gentle, modest, and golfer-friendly: the 6500 yard total length, the only slightly undulating terrain, the forgiving greens, the wide fairways, the lack of water and fairway bunkers, and the healthy but simply maintained turf; and the site itself (as with many farmer's fields in the area) is surrounded by a single row trees - originally there as a wind shield/breaker to protect the crops.
And because all of these elements work together and are mutually supportive, the course has a character all of its own -- an open, friendly, and humble character that welcomes golfers young and old, beginners or veterans, to enjoy the game. Now, that type of character may not get high rankings or win awards, but it is character nonetheless, and of a kind that serves the community very well.
In fact, the only element out of character for this particular course are the new trees they planted a few years ago between golf holes and/or to cut off potential shortcuts -- and they are out of character precisely because, in trying to be a tougher test and make the course look/feel less like a farmer's field, these new trees fly in the face of what the course actually is.
It's not that man who wears bell-bottoms and a fedora and a leather jacket and sneakers who has/demonstrates character; he demonstrates instead that he's an insecure and attention-hungry slave to fashion. It's the construction worker un-self consciously getting off the bus in his workboots and stained pants who seems to have character; and it's the woman on a hot day with no makeup on and her hair pulled back in a bandana wearing a loose and colouful summer dress that flutters in the wind and a pair of sandals who seems comfortable with herself and at ease -- the true marks of character
Peter