I'll chime in as a person who has a so-called "circle of friends" and burst the bubble right here:
I see that 'circle' about once a year, at my husband's and my birthday party. We invite them, we all have a good time, and some of them, I might see once before the following October, and likely at one of their parties. They have other, closer friends, and I have friends I see more often as well. I love seeing them, but to be truthful, I have just three 'regular' girlfriends... and we all do not usually hang out together.
I'm very introverted, so one-on-one relationships feed my soul so much better. I have two 'older' friendships that are born of circumstance: one is my girlfriend from high school, where we were misfits with wacky families; one I met in my very early twenties, where we were misfits at our church, so to speak. My last, newest girlfriend is a mom I just 'clicked' with, and you know, it took a long time for this last relationship to deepen outside of the mom's group (which I rarely participate in anymore...too busy).
A big part of what made all of these relationships happen was follow-through on my part. It meant calling just to see what's up (which isn't easy for me at the onset of a new relationship) and really putting myself out there.
There's also the flip side, which is to say that I've met many other women I thought would be neat friends and suffice it to say, it didn't pan out. Within my 'circle' I previously discussed, there are only one or two couples that make an effort to get together aside from the once-a-year events. A lot of the reason we get together is because I make it happen, make the calls, schedule, reschedule if need be...
and so this is what I will hand back to you: if you like a particular person, try taking it one step further. Keep calling for those good playdates and then try to expand them for a picnic outing, a hike or some other common interest. Don't just mention that you want to get together for coffee or go out for a movie or drink, call and ask specifically "Hey do you want to go and get a cup of tea on Tuesday night" or 'Brunch on Sunday' etc. Also, most of my mom's group acquaintances do have a harder time getting out, which is why you might see something that looks like a circle of friends... but behind that was an email that went out three weeks before and even still, not everyone could make it... do you get what I'm saying? It's mostly about circumstances, recognizing opportunities and making an effort for it to happen.
For what it's worth, I wouldn't focus negatively on the descriptor of 'being the girl she plays sports with'...it sounds like this person really appreciates that she can do this with you, and it gives more information than 'oh, she's a friend'. But I can understand why it stings a bit. Just wanted to say the grass isn't always greener...