Any tool where they specifically tell you not to leave it plugged in is not a tool I'm going to buy...having seen wiring from some goods made where I suspect this one is made, they are not big on safety...
I actually use the Grobet/Vigor unit they show on the page - their part FB-119.
To put as simply as I can how demagnetizers work, we need some simple descriptions. You need to imagine that a piece of ferrous metal is made up of particles that all have +ve and -ve poles, so tiny little magnets. When particles all have their poles oriented in the same direction, the part is magnetized and creates a field that can attract ferrous metals. A demagnetizer works by randomizing the orientation of the poles in the particles inside the metal. This is done by using a strong alternating field to start the poles swinging inside the metal - ideally all in unison at first. Then as the field gets weaker, poles will stop flipping back and forth a little at a time, and when done they will all be randomly oriented. When that happens, the part is no longer magnetized.
There are 2 styles of demagnetizers you can buy. The one I use is an instantaneous style, you there is a capacitor that is discharged when you press the button, and it creates an alternating field that decays over time. You hold the watch above the unit, press the button, wait for the cap to recharge (couple of seconds) then do it a couple more times. The second type can be flat or tunnel shaped, and with that style you place the piece to be demagnetized in or on the unit, then press and hold the power button. You keep holding it while slowly moving the parts away from the field until you can't get it any further away (so the length of your arm) before letting go of the button. With this type, if used improperly, and you press and release the button with the piece still in the strong field, you will magnetize the part further, rather than demagnetizing it. The tunnel style typically have the strongest fields.
I would not recommend buying a demagnetizer that was not designed for watches - such as a bulk tape eraser or something. If the field is too strong, you can damage the watch by tangling up the balance spring.
You can detect magnetism with a test, but the compass test is often not done right. First you need to understand that a compass has a magnet in it, or it would not be drawn to magnetic North. So just putting a lump of metal on the table (watch) and putting a magnet near it, and saying because the needle moves the watch is magnetized...well that's not really the case. I don't know of many watchmakers who use this type of test anymore, but from some old timers who did use it they told me they would sit the very small compass over the balance and watch for the needle to move at the same rate as the balance was beating.
I typically use the timing machine and observation of the balance spring, but if you can find a Gauss meter that could also work. I have an old style one that I was given by my workplace when they upgraded...in the background here:
Oh, and since quartz watches have magnets in them (the rotor is a permanent magnet), subjecting them to strong magnetic fields is not a good idea.
Cheers, Al