Sorry Zelator - a witch and a magician, when referencing modern practitioners, are the same thing.
The definition of "witch" goes as such;
"a person, now especially a woman, who professes or is supposed to practice magic, especially black magic or the black art"
Now if we were to ask literally any serious practitioner who uses the term "witch" we would discover that the old social prejudices about magick found in this word ("especially a woman" and "especially black magic or the black art") are left out and so we are left only with "a person who professes or is supposed to practice magic."
Let's look at the definition of "magician" now, excluding those definitions which specifically reference stage magic;
"a person who is skilled in magic"
... so a person who professes to use magic and a person who is skilled with magic.
Sounds pretty damn similar, don't you think?
In modern usage the only difference is which groups prefer which terms - something no more significant than preferring the term "dog" over "canine" in daily usage.
The only reason they are separate words is because English is primarily of Germanic origins but was heavily influenced by 11th century invasion of the Normans, who introduced the linguistic precursor to French. "Witch" comes from our Germanic side, "Magician" comes from our French side.
As for your pointing out that "when you call upon a demon, you NEED a magic wand," congratulations - you've observed a tool required for a specific magickal working. If we are to take the logic that assumes that a wand is therefore necessary for all magicians, every tool, sigil, stone, plant, incense, oil, and article of clothing used in a specific ritual is suddenly necessary.