No. A magick wand is for magic…in other words, its purpose is to strengthen spells (but basically, most that I know don’t bother with it…it’s more often used with magic “kits” for kids).
A healing wand is used for healing…while some people consider them the same, they’re not. Magic is generalized…healing is specialized.
This is an interesting question, but not one that can be answered completely outside of a particular magical school or oathbound tradition.
Wands have a specific use and purpose, like a hammer has a number uses, and a screwdriver has a number of uses, but they very rarely overlap.
Technically speaking you could use a regular wand for both purposes but the copper/crystal wand for fewer things.
Every magickal tool you use has to have a purpose, the things it is made of determine what kind of a thing it is. This depends a lot on your magickal school. Most people would consider copper to be associated with Venus/Aphrodite or feminine at the least, it is also a good conductor. So this material would probably be more for love, attraction or sexuality than healing per se. Quartz is a good director of energy so that makes sense.
If the purpose is “healing” there are a lot more appropriate materials, and I would say most of those are wood/plant based. An “asclepius staff” comes to mind. Each wood and plant also has correspondences that would make it more or less appropriate to your work.
Finally each tool has a use, it has it’s own properties (gender, element, activity, etc.) that is more or less appropriate to a particular type of magickal operation.
You may have different associations depending on how you were taught or what you believe etc. (Is the metal mercury water or air for example).
Ultimately you have to have a reason and it has to make sense.
If you look at people’s altar setups online for example you will often see all kinds of bric-a-brac on there. Forty candles, 5 chalices, stones, necklaces etc. Everything that you use in a magickal operation should be there for a good reason. I personally might use a wand but it won’t be on my altar if I’m not using it. If I had a statue of Hephaestus (for example) on my altar I would probably not have it there for a healing operation (he was lame after all).
If you don’t follow a particular magickal school or tradition, your best bet is to use whatever feels most right. Try not to have 100 wands for 100 purposes, none of them will get particularly powerful through use.