Neil, "Weedeater" is what we usually call them, too, although that's actually a brand name. I guess they were the first to make them. I think the generic name is "string trimmer" and I've had 4 of the type you mentioned; two electric and two gasoline. But with getting older, and having carpal tunnel syndrome surgery on both hands, I bought the DR trimmer/mower just like Harv has; less vibration for your hands, no weight to carry around, electric start, and more power. And like Harv, I have the "Beaver Blade" so I can take the string head off, put that blade on, and cut down little trees up to about 4.5" in diameter. And they call it a "trimmer/mower" presumably because you can cut at a more consistent height, so you can actually use it as a lawnmower if you want to.
And the DR trimmer/mower (by Country Home Products in Vermont) was the first string trimmer of this type I ever heard of (don't know for sure they were first), but now lots of other companies have also copied the basic design, so you can buy them lots of places.