Dario,
You are right, you cannot reuse copper wire. Copper wire "work hardens". Which means that it gets stiffer the more you bend it. Which is precisely why it works so well!
For those new to copper wire:
Annealing (heating the wire until it glows red) realigns the copper atoms in the wire. Which makes it easy to bend. When bent, the atoms form a crystalized structure, which is stronger and stiffer. (I am not a chemist/physicist, I'm just relaying a layman's understanding of why it works.)
What this means for us is fresh, annealed copper wire is easy to bend and apply to the tree. When cutting off a piece of wire from the roll, try not to straighten it out, and as you work with it, only bend the part that is being wrapped around the branch at that time. I often see beginners cut a piece of wire, maybe about 18 inches long, and then immediately straighten it out! And then start to stick it in the tree. Don't do that! Straightening it out stiffens it. Learn to work with it while it still has the coil shape to it. It will be easier to bend.
As you bend the wire around the branch, it stiffens as we make the coils. Once we have wired the branch, we use the wire to shape the branch into the proper growing position. When we do that, we bend the wire again! So, it stiffens even more. Each successive bending stiffens the wire a little bit more. So when we are through with our wiring and styling, the wire is pretty stiff, so it holds it's position. Which is what we want.
Aluminum doesn't work harden.
I teach wiring at Plant City Bonsai. Beginners are surprised to learn how easy annealed copper wire is to apply. (I think they may have tried to use old electrical wire, and it hasn't been annealed.)
Annealed copper, by the way, often has black carbon on it from the firing process. When it's wrapped around a branch, that look goes away, and the wire looks almost red. Not the bright shiny (almost gold) color we associate with "copper", but a dull dark red. In a week or two, that oxidizes to a dull brown.
One more thing...
Removing copper wire:
The old school teaching was to not unwind it, but cut it off. (Because it's stiff.) That is still true for gauges 12 and larger. (Gauges for copper wire are small numbers for big wire, big numbers for small wire.) So gauge 12, 10, 8, etc get really stiff. Really hard to unwind. But most of our wiring will be 12, 14, and 16. (Sometimes gauge 12 can be unwound, sometimes not.) But those smaller gauges should be removed by unwinding. Grab the tip end of the wire with pliers, and unwind the wire. Back to the "anchor" point. Then (holding the wire at the anchor point), grab the other end of the wire at the tip with the pliers, and unwind back towards the anchor point.
Doing it this way removes all the wire. If you cut it off in little pieces, you can accidently leave little pieces of wire in the tree. Especially in the crotches, which can get stuck in the tree permanently. We don't want that.
You can't reuse those twisty old wires. Save them in a box and take them to the recycle place, or sell to a copper buyer.