What is the rail made out of on your balcony? In my old apartment it was
all wood. Incidentally, my cats went out there all the time and never
even jumped up to the railing, never tried to jump down to the ground. It
was only one story up from the ground, too. Marley used to lean out but
he never jumped. I suppose I was lucky. I lived there 4 years, the whole
time he was there with me.

As for material being visable, the fence material I used is designed to
be invisible. I think if you attach it to rods that lean inward and go up
about 2' or so, they won't even bother to try to get over it.
Sort of like this:

 /
/

Then use the same mesh/fence material to enclose the bottom rail.