Is there such as thing as a one horse closed sleigh?

Sure, why not.
And why? For lyrical reasons.
Maybe for completeness’ sake.
“It is a car” just does not sound as well/tell you as much as “It is range Rover with a 500 horsepower, supercharged 5 litre v8 engine.”

Sure. In some areas like Russia they’d use an open or topless sleigh & furs, but topped sleighs were obviously warmer.

Only when the horsey sleigh union vote to go on strike, usually over hay pay or preferential treatment to those jumped up reindeer.

An open sleigh may also mean one with no overhead cover. Could be completely enclosed but have never seen or heard reference to one. There must have been some variations, like an enclosed stagecoach.

Closed sleighs weighted considerably more – walls, roof, windows, etc., and required two horses to pull the weight.

@elbanditoroso GA, and blocked your view of the countryside as you travelled.

“Open” may be the redundant modifier, since it seems rare to find a “sleigh” that isn’t already open (including Santa’s, of course). But “one-horse” is a necessary modifier, because there are many different multi-horse sleigh arrangements, including timber, lumber and other heavy cargo sleighs, as well as the one-horse variety (optional two-horse varieties as well) for light personnel carriage.

In that case I suppose that “one-horse open sleigh” was used as much for the meter and rhythm as it was to allow people who had zero experience with sleighs of any kind (such as most of us) to form a mental image picture of what it might look like – and how it might feel.