I want to put a hand rail along a steep portion of a trail in the woods, so old folks like me can get along it more safely/easier. I can lash the horizontal pole to a vertical tree, using a steel wire. In a few years the tree will grow around it, making a weak point in the tree and hurting it's grow as well. I could use a plastic coated steel cable, which is thicker than the wire, but I'm afraid it will prevent growth as well, just more slowly. I don't think nails or bolts in the tree will do it well, true? Could I put a protection under the cable, so that the tree would continue to grow inside the cable protection? Is there another way to attach the railing without damaging the tree?

You might consider just putting a pole in the ground for that point, as using the tree will indeed cause some harm to it. If the wire goes right around the tree the tree will eventually grow over it to the point the tree will be basically 'ringbarked' where no food can reach the roots from the leaves where it is produced. Then you have a hazard on your hands.

Yes, 'ringing' the tree with wire is not a good plan. Esp. these trees, that are about 6-8 inches dia.

My solution, semi-approved by a forester type person, is to (a) use plastic coated cable, (b) put sticks vertically between the tree and cable, so the cable never touches the tree bark, and (c) promise to adjust the whole thing avery 2-3 years. Turned out getting the cable around the tree, and tight, was a task and a half even for 2 people. But the horizontal pole is tight and supports people walking down the steps.

Now I have to come back and adjust it every 2-3 years.