I cannot tell how sturdy your posts are from the picture. I can tell how thin they are by the relative size of the insulators. The posts will hold up your fence wires but they won't tolerate much lateral tension without shiftingand loosening. They won't be as sturdy in the ground as thicker posts. You will also find it difficult to hammer in insulators in a thin post as the post will flex quite a lot. I would also be worried about the thinner posts splitting when you nail or screw in your insulators and then not having enough lateral surface area to place the new nails while still keeping your fencing wire fairly level.
That said, you could still use them if you put in a few extra thicker posts every 20-30 feet or so. With some brands of electric rope or tape, posts can be up to 60 feet apart. This is OK as long as the area is perfectly flat, it if you have any rise or fall of the land, you will need posts in between. This is where your 3" posts could be used.
As far as the fencing strands themselves, there will be pros and cons to each. The high tensile is terrible to work with and it will cut a horse badly if they get tangled in it. The electric rope will injure a horse as well, but it is much easier to put up properly. Electric tape looks nice when taught, but takes a beating in the wind and requires regular tensioning. You can also get a polymer covered electric wire that looks nice, but it stretches out of shape if a horse leans on it, and like the others will cause injury of the horse gets tangled. Never, ever tie a horse to a thinner post (less than 6") as they can pull a 6" post out of the ground quite easily.
So the key is this: Use good posts to support your fence, especially in the corners, no matter what you choose. More strands is better, especially at the bottom as this is where horses are more likely to out a foot through. Electrifying a lower strand will discourage this. Use the best insulators you can get. I prefer the screw in type as nails work themselves out. Keep your fence properly tensioned with the proper parts and check it regularly. Rope doesn't need to be as tight and it will stretch in the cold. Wire and coated wire must be very tight. Buy an adequate electric discharge unit for the distance you a fencing. Buy a tester and test the fence regularly. Ground your fence properly using the proper materials. Don't make substitutions here that won't work as well.
Brand of fencing isn't as important as proper installation. Check the cost, warranties, visibility, tensile strength and conductivity if you want to compare vrands, but no matter what, use the appropriate parts and purchase something you can easily maintain yourself and buy a little extra for repairs, etc.
Not sure what other advice I can give you, other than "happy building"!