Dr William Lee Dawson - What Christian name did he use?

I've always referred to my great, great, great grandfather by using both his christian names William and Lee. This was probably to distinguish between father and son, William Henry Dawson. 

Dr William Lee Dawson 1819-1871

William Lee was named after his maternal grandparents. His grandfather was William Giles and his grandmother, Catherine Lee.

The Lee branch of this family has been well documented.  William Lee's maternal grandmother Catherine Lee was the daughter of the Rev Michael Lee, the son of Richard Lee, the son of William Lee who was the son of Emanuell Lee. 

The Lee family are mentioned in Rev George Hill's book The Plantation in Ulster. Others are referenced in Evelyn P Shirley's book The History of the County of Monaghan. The Rev Michael Lee graduated in 1726 with a BA from Trinity College, Dublin. 1

Now I'm wondering if my great, great, great grandfather went by the name of Lee, not William.

The only evidence I have for this is the death notice of his wife Emma in East Melbourne in 1885 but it is enough to make me wonder. It says she is the relict of W. Lee Dawson. 

The Argus (Melbourne, Vic. : 1848 - 1957) 29 July 1885, p. 1

William Lee, were you William or Lee?

1. http://homepage.eircom.net/~leeea/EmanuellCavan.htm