Honestly I grew up with a "horsey" family and now live near lots of farmers who rent out fields for horses, not once have I see a horse having a kip, while standing up and with its eyes open. Far more common to see one flat out on the stable floor with the barn cat somewhere nearby! Granted you do see a stubborn one at the County Show with its head dropped and eyes shut, but that's probably the short bursts of sleep/dozing that your quote refers to.
Meercats on the other hand... seeing them huddled together under a heat lamp in a circle; they look like they belong in the smoking shelter at a JD Wetherspoons.