The two-ring AF-D, often called "Tripod Mount", is the oldest one you should even consider. It focuses fast for a screw drive lens. The other pre-AFS 80-200 f2.8 lenses were real auto-focus speed dogs. But, keep in mind that with screw drive lenses, focus speed is to a great degree dependent on the motor in the camera. This means that the lens will focus much faster with a professional camera like the D4 (which has a very robust motor) than it will with a D7000 (which has a less powerful motor). An AFS lens doesn't really have that issue, as the motor is in the lens, and the focus speed is limited only by the lock-on speed of the camera.
I am a sports shooter, and I am also considering which XX-200 f2.8 to buy. My budget also will never allow for a 70-200 VRII for my hobby, although this is the best XX-200 f2.8 lens in the world, today. So, like you, I have considered the older 80-200 f2.8's. I have decided to buy the AFS 80-200 f2.8 used, when my budget allows. At KEH, it sells for about $1,200 used. I find that with my D7000, there is a noticeable difference in focus speed and accuracy between my AFS lenses and my older AF-D screw drive lenses when shooting action.
That said, I'm pretty certain that a good photographer, with good technique, could use the AF-D Tripod Mount version and get great results shooting sports. For me, focus accuracy is the single greatest determiner of good sports results - and I need all the help I can afford. My cameras seem to nail white balance and exposure pretty much dead-on almost all the time. The majority of throw-aways for me are the result of focus errors - either loss of focus lock (my fault), improper target acquisition (also my fault), or slow target acquisition and lock-on (the camera/lens' fault primarily). As such, I want the fastest focusing lens I can afford. If you can swing the cost of the AFS version (and if you can find one, as there were far fewer of them sold than the AF-D tripod mount version), I'd recommend that version for a sports lens.
BTW, the used AFS 70-200 VRI sells for about $1,600. It is reputed to be a touch soft in the corners wide open on FX and film. But, on a DX camera like the D7000, that should not be an issue. It is a generation newer than the AFS 80-200, and it has VR (useless for sports, but someday you might want to shoot something else). Is it worth $400 more than the AFS 80-200 f2.8? Only you can make that call.