There is no specific definition. It usually means mice with capabilities tailored for gaming:
-high DPI
-on-the-fly DPI
-high report rate
-(can work during) high acceleration
-macros
-customizable weight
Some manufacturers also take it as an excuse to charge more.
Be careful if you open up your Trackman. I lost one of the ruby bearings while cleaning mine 
Vote for $23 LogitechTrackman Marble.
If the OP is hovering his hand, then he doesn’t know how to use a centered trackball. The base of the palm rests on the base of the device. The fingers rest on the ball.
I’ve had to use a mouse at work recently, and would have to lift it to reposition it every 10 minutes or so. That would destroy my concentration on what I was doing. Maybe I could have upped the sensitivity without impairing it’s ability to do fine detail, but it’s just not an issue with the trackball.
Tried a thumb trackball for the first time last month. Presumably people get used to them, but my immediate reaction was horror. That’s not a feeling I’ve got from mice, digitizing pads, touchpads, or centered trackballs. I gave it two tries that lasted about 5 minutes each, and then used the company mouse until I could buy myself a Marble trackball.
A thing about just upping a mouse’s sensitivity: Sometimes I’ll drag things slow and far. A mouse has a lot of trouble with that.
I might be using Photoshop, and when zoomed in 500%, cut out some trees and drag them the width of several screens. With trackball it’s no problem, I just keep spinning the ball till I get where I’m going. Might take two seconds.
In InDesign I might drag a photo up to the previous page, and the program might balk for five seconds. If I keep spinning the trackball, the photo will get there without drama. If I’m using a mouse I’ll have to lift it and reposition it multiple times.
“Helloo… computer?”
—Scotty
I’ve never used a trackball that wasn’t far less precise than a mouse. It’s much easier to be precise moving your whole hand than just moving one finger. With a mouse, I can move exactly where I want to go. With a trackball, I have to swipe the ball multiple times to get anywhere, unless I have the precision set horribly low.
I also can’t figure out any reason you would actually need a high level of precision where a tablet would not be superior. You’d have to be doing some type of graphic work. Otherwise, the UI elements you are using will be large enough that such precision is unnecessary.
My mouse moves less than an inch to cover my whole screen. With the OP’s setup, that’s less than 3 inches. I don’t see how that’s harder than using a trackball and swiping. I find that as annoying as using a trackpad.
First, let me say this: I have used a thumb-ball mouse, and I really liked it. I am happy that the OP found true hardware love, and I respect his choice. However, in response to this:
Consider the following: All professional gamers who play real-time strategy games use traditional mice. All of them. And they can all click or draw a box quickly and precisely anywhere on screen from anywhere else on screen. How do they do this? Lots of practice, and fine-tuning their mouse weight and sensitivity to find the perfect settings.
My point is this: the standard mouse may not be the most intuitive or the most ergonomic approach for everyone, but its responsiveness and simplicity of design give it a very high skill ceiling. To describe it as “inefficient” indicates that the OP has not unlocked its true potential.
I am in GIS. I make maps and do a lot of data update where precision is very important. I have zero problem doing that with a mouse on my 2-20 inch screens. The movement of my mouse is perhaps 2 inches to cover both screens. Your two foot circle comment is silly.
:shrug: To each his own. I absolutely loathe trackballs. To the point that I will take a wireless mouse with me if I have to do anything on someone else’s workstation
Wondering if maybe the trackball people have arthritis, or something else which interferes with fine motor control.
This thread isn’t really about mice. It’s like saying my Segway doesn’t have enough battery for a tour of my mansion, woe is me.
My motor control is just fine, thank you, but I started getting twinges of RSI in my hands so I switched away from mice and I’ve had nary a twinge since.
In essence, yes, ‘gaming mouse’ means ‘nicer mouse’. A proper gaming mouse will allow higher precision, will often have more buttons (which you can program and use, or ignore), will have the previously discussed on-the-fly DPI adjustment, and may be built for heavier use/abuse. Everything is improved over your average cubicle-dweller’s company-provided mouse. Even if I didn’t game, I’d buy a ‘gaming mouse’.
If you can, go to a place that lets you try mice before you buy them. Best Buy used to have them on display, so you could try moving them around and seeing which one seemed comfortable. Their display wasn’t exactly designed for that - more to let you hold it and look at it - but it’s better than nothing, which is what most others offer.
If you can’t try any mice before you buy, I might suggest the Logitech G9x. It has two shells which appeal to different people depending how they grip their mouse (I use a fingertip grip myself.) I bought it thinking I’d like the small shell, but when I got it, I found out I liked the bigger shell. Had it been most other mice, I’d have been stuck with a mouse that was too small to be comfortable for me, or I’d have had to return it.
Then when you get a mouse, you need to look into a mousepad. 
I’m a mouse user, but my wife dearly loves her wired Logitech thumb trackball. I keep it running for her, and have repaired the cable and replaced the switches once already.
Doug Engelbart, the inventor of the computer mouse, died today. Can anyone verify Amateur Barbarian’s whereabouts earlier today?
What do you mean, us? I was right over here.
I paid $25 for a Logitech wireless mouse and keyboard almost two years ago, and they’re still in perfect working order. My mouse is set to the highest sensitivity, and I barely have to move my hand to have the cursor cross the entire screen. To make the mouse and keyboard connect, I just plug in the USB stick; I’d estimate the whole process takes roughly fifteen seconds.
People talking about bad wireless connections and cursor drag times are speaking a foreign language to me.
Extensive use of a mouse gives me tendonitis, clicking the left mouse button with my index finger causes particular problems. I’ve used a number of trackballs.
This logitech model I found unacceptable because it didn’t have a scroll wheel. I like scroll wheels and those little buttons didn’t cut it.
I went through 2 or 3 of these numbers, back when it was still made. This Logitech ball was ergonomically friendly, not especially cheap or durable and wireless. Wireless is a disadvantage as it involves batteries and is another thing which can break. Hint: place the receiver within six inches of the unit. 3 stars. (Now available for $350: OMG).
This is my current favorite 4 star device, purchased in Jan 2010: $30 Kensington Orbit Trackball with Scroll Ring. I click with my thumb. Wired. Reasonably durable. Cheap. Workable scroll ring. All it lacks is a middle button. A poster at another message board said that the scroll ring had a counterintuitive orientation (clockwise sends the page down) but I haven’t had that experience. Also cat hairs would clog the device up. I haven’t had that experience either: I’ve only dismantled it for cleaning once after about 3 years. I do remove the ball and tap it on a table periodically. Recommended.
Thing about a mouse is it is not where it was a moment ago. The trackball is always in the same place. Your pointing hand does have to jump to the keyboard from time to time, unless you are not a touch-typist, I find it of great value to have a pointing device that stays put.
Of course, the mouse people are getting more exercise. Perhaps that is a good thing, though not everything that does not kill you will make you stronger.
I second the recommendation. I too use one.
Switch hands. I had to become ambimousetrous.
This is actually not a good idea as you end up with issues in both hands - I speak from experience.