I was thinking about the lenses, but also about your body choice.  Bill has a 5DIII, so I asked him about it to make sure I'm not off track.  He says if you're shooting sports with a telephoto, you probably don't want the 5D.  It's a full-frame camera, which cuts your telephoto range from 1.6 to 1.0.  That means with any telephoto lens you'll see farther with your 60D than you will with a 5D.  Instead of the 5DIII, you might want to think about moving to a higher-end crop frame body.  (I hear they're coming out with a new 7D and/or 70D in August, which will have the new high ISO chip.) 

If you decide on the 5DIII, you'll need a longer telephoto lens to compensate for the full frame format.

I've got the 70-200 L-series, and the thing I like about it over the 70-300 is that the f-stop range goes from 4.0 to ... whatever.  20?  I can't recall.  The 70-300 is limited to 4.0 to 5.6.  That doesn't give you a lot of latitude.  Also, the longer the telephoto, the softer your images are going to be, so you'll sacrafice a bit of sharpness to get that extra distance.

When we were at Bryce Canyon last month I was happy to have the 70-200 and the full f-stop range.  There are a lot of sunny spots, but also a lot of shady spots, and it was great to have a wider range of choices.

The 70-300 is a little bit lighter in weight than the 70-200.  On the surface that sounds like an advantage, but it means you've got less glass ... and less glass is the reason for the softness of the images.  Better glass equals better sharpness.

I hope that helps with your decision making process and doesn't muddy things up too much!  LOL!

"If your pictures aren't good enough, you're not close enough."  ...  Robert Capa