I’ve read both, but never actually heard either used.
For what it’s worth, and since we seem to be visiting the euphemism, anyway, “Ode to the Four Letter Words” dates back to at least the 1940s and offers this verse:
That suggests the “dog” version was at least applicable, if not particular, to urination at that point.
Horse. Or:
“I’ve gotta drain my lizard.”
I have to go return some videotapes.
These all mean I’m headed to the restroom:
“… to see a man about a horse.”
“… to shake the weasel.”
“… to shake the hand of Abraham Lincoln.”
Wikipedia says “dog” was first (and samclem’s cite agrees). It also says “The original non-facetious meaning was probably to place or settle a bet on a race, thus dogs or horses”.
I’ve never heard anyone use either version of the phrase except myself, and I’m pretty sure I learned it from the Dope.
And when I did use it, it was at my old job as a fast food manager, as a way of telling the shift manager “I’m making a bank run, but for security purposes I’m not supposed to say I’m making a bank run”.
“Horse” here. Never heard the “dog” variation, to my knowledge.
In the UK, the phrase “I’m going to see a man about a dog” means " Ive got things to do and I don’t need to tell you about it".
Exactly as the OP said.
That’s what I mean when I say it. The only thing I mean. I’m from Philly and have been in Texas only a couple times.
In the late Forties, Ralph Moody began writing a book called Little Britches. It was set in Colorado, I believe. I didn’t hear of it until about the mid 1950’s.
The father in the story used the expression, “I’m going to see a man about a dog.” He was actually going to give the principal a piece of his mind about how his son had been treated by the principal at school. His use of the phrase was a way of saying, “I’m not telling where I’m going nor why.”
That’s the only way I’ve heard the expression used.
Horse, always. I use that myself, regularly.
Dog - never heard that used.
There’s also: I’m taking the Cleveland Browns to the Super Bowl. Happens every day.
Say what? 
Yeah, I was wondering about that one, too.
I wonder if it comes from the Confederates…?
Also in Atlanta, also always heard horse. I’ll third the American Psycho video tapes quote as my personal favorite.
I see I was ninja’d on “Gotta take the Browns to the Super Bowl”.