electrobraid horse fencing?

Height, in my opinion, depends if your horse likes to jump. Mine don't, and they will all stay in something as low as belly height.

Livestock fencing is good, but you want to have a top line on it, because they will reach over the top of it to eat the "greener" grass.

The true factor of why animals will push through one fence but not another isn't as much the type of fence but what is attatched to it.

A good fencer is a must especially if you have a horse that tests fences, granted a good fencer with a super cheap poly wire isn't going to do as much because it's not going to be quite as connected as a solid wire

The person asking about using poly fence as a top and hard wire on the bottom, I personally would switch it to the hardwire on top, or just use hard wire for the perimiter and then use the poly wires to section off areas for grazing/regrowth

Your horse knows exactly where the wire is. I used to watch my graze and his ear would be within half an inch of the single strand - nerve wracking to watch. I've watched the horse run and slide on wet grass thinking that's it, they're going thro but both always managed to get stopped in time, close, but not touching. Horses seem to have the ability to map their territory as has been noted with blind horses.

that's why I wanted to put a wire over the mesh so he wouldnt dare lean over it.

I put a single line around the perimeter of my field attached to the inside of the other fence and chest high. It has done away with all the leaning over to eat the neighbor's hay. To train the horses to electric fence I ran a twenty foot piece across an opening and put grain and hay on the other side then turned it on. It only took about twice for them to figure out where the sting came from. Didn't have a bit of trouble until this mare came that had been kept in bad fencing and she always tested to see if it was on.

Is electric fencing by itself actually safe for horses?

Yes, it is safe.

Our fence consistent of 4 strands of nautical grade electric rope (7mm, not the thin stuff) hung on treated wooden posts 16'apart with screw in insulators. The top and third strand are electrified. We have used this and only this for the past 2 1/2 years to keep our four horses in with very few problems. I had one strand come undone this week partly because the fastener was a bit loose (cold shrinkage) and partly because my draft cross was pawing at the bottom wire to get at a new mare across the fence. The rope didn't break, it simply came undone at the joiner, but held at the next insulator. The three strands above it were fine. This was also the non-electrified strand. I'm pretty sure all my horses knows which strands are hot and which are not as they get very close to the fence but don't often get zapped.

This type of fence works well in all weather and is easy to maintain. It is important however to use a good charging unit, ground it properly and select good posts for support. I notice our fence does tend to expand and loosen a bit in the cold, so I do check it regularly in the winter. For reference, our temperatures range from +35C in the summer and can go down to -45C in the winter. The fence takes this, no problem. Even with 2 feet of snow on the ground, the conduction is good at -40 (as I found out the day before last).

Ok I was thinking about doing something like this:

How thick should the wooden posts be for mesh?

That's what I have, but the hot wire on top of the posts.
How thick will depend on what distance they will be.
If you want them to completely stay away from the fence, run another strand hot inside the mesh, about chest height.