How they're constructed makes way more of a difference than the color alone. Your first picture, vinyl posts with pre-drilled holes to accept vinyl rails, is expensive crap in my opinion. They're nice-looking when you first install them, but don't hold up after a couple years of UV rays, horses pushing against them, the ground freezing, thawing, getting waterlogged, etc. My neighbor put up one like that, and it wasn't even a year before the posts were leaning or bending every which way, causing the whole fence to look really bad, and creating gaps between the rails so the animals could get out easily if they pushed against it.
The second picture looks better, it looks like treated wood round fence posts painted white, with some sort of flexible poly rail product attached to it. The only thing that's objectionable here, is that the rail product is attached all around the outside of the posts. Rails should always be attached to the inside, so that animals can't detach it by pushing against it. For continuous, flexible rail products that have to be stretched, such as wire field or no-climb fencing, or creative options like surplus fire hose, the best way to attach it is to brace the corner posts and roll the fencing on the outside of those, but roll it back to attach to the inside of the posts along the straight stretches.
The third picture looks just like the first, except black, so just as bad.
The last picture looks the best, in my opinion. Treated wooden round fence posts, painted black, with the rails attached to the inside.
The only reason color makes a difference depends on the climate where you live. Where I live here in the Pacific Northwest, we get a lot of rain and moisture hangs around forever, so white vinyl fences quickly get green and algae-covered, and water soaking into the ground for most of the year moves fence posts that are too light or not concreted in. There's a reason why all the fancy racehorse places in Kentucky have black fences - there's less maintenance needed, plus they're easier to see if there's snow.
For safe and long-lasting fencing no matter where you live:
Use heavy posts, like treated wood, not those light vinyl things. Put 1/3 of the posts under ground, like if you want a 4-foot fence, use 6-foot posts and bury them 2-feet deep.
Always concrete the corner posts and brace them. If you live in a place where the ground is mostly sand or silt, concrete all your posts, but if you live where it's rocky or gravelly you can get away with only concreting the corners and using metal T-posts in between. Distance between posts should be closer (like 8 feet apart) in sandy or silty ground, you can get away with up to 16 feet in more solid ground. Whatever you use for rails, install them on the inside, except for a continuous, stretchable product, which goes around the outside of the corner posts, then inside all the other posts.
Once you have a safe, solid and long-lasting fence, paint it whatever color you want!
