I was just about to respond about fanning. After giving it a little more thought, I think the term makes sense, and your description is pretty good: if you're playing lazy 8ths with the stick while playing lazy 8ths with your foot, and if done such that the stick hits the hats at or around the same time as the foot closes them, there's a kind of airy and sloshy thing happening while still maintaining some amount of articulation. Is this what you're describing?
The best example of this I can think of is on Abbey Road during The End right after the drum solo, where Ringo uses this kind of riding technique right through to where everyone stops and it's just piano 8ths.
I'm not including the obvious accented hat openings that close on many of the 2 & 4s thru that section, though. To me, those are just run-of-the-mill disco barks (albeit tasty mellow barks done within the context the effect described).
In any case, I'm not hearing that effect on Take the Money and Run - to me those are barks because the top cymbal is far enough away from the bottom cymbal for long enough to open up clearly and is then cleanly choked off when the cymbals come back together. But maybe you're talking about something going on between those that isn't obvious to me on my crappy computer speakers?
I hereby rescind my ban on the word "fan" for drumming applications
... as you were, men - smoke 'em if you got 'em!
BTW, I'm digging this discussion about tension and stiffness in the metal and its effect on how cymbals play and sound. Paragon hats, according to Sabian, are medium weight both top and bottom, so if anything they should be lighter and easier to play than New Beats, Ks, A/K Mastersounds (though not A Custom Mastersounds which are medium-thin over medium), Evolutions, Stage Hats, AA/HH Regular (or medium) hats, et al, ... all of which use a medium over a heavy.
I've played Paragons before, too, and wasn't particularly fond of them because they just didn't seem all that responsive, despite their "lighter weight" (but I think Sabian may be playing loose and fast with those terms, tbh). There is definitely something else at work here.
... which leads me to this bit of forum flaming (kidding!) - as much as I like Sabian cymbals, there is something in the metal they produce and work, and its present in every line they make ... I don't know how to describe it, but to me they sound mellower across the board than their closest competitor (Zildjian). My pet theory is that Sabian metal has less tension than Zildjian such that even my Ks tend to be bighter than my AAXs. I had a pair of AAX Stage Hats, which were kinda cool and not bad to play, but they didn't seem to produce all the frequencies that my very typical New Beats do, and were surprisingly mellow for hats that are among the brightest in their entire catalog. Zildjian gets hated on for being harsh, and I often agree with this criticism, but at the end of the day, they're more even across the frequency spectrum even though many of those overtones seemingly conflict, giving them their harsher sound (especially with the A series). Sabian manages to avoid those conflicting overtones simply by not having as many there to begin with). This is not necessarily a bad thing and is part of what gives Sabian the charm they have, but it seems to be a result of their cymbals being somehow softer (lower alloy tension) than Zildjians.
Anyway, all this to say that a subtle technique like "fanning" won't be as effective if the cymbals in question aren't producing the full array of frequencies and/or overtones.