Personally I don't like the sound of any of those sound clips on that website, I would've spent more time finding the best mic position for a recording, but since it's mainly an article about live mics, rather than recording mics I suppose they had to put the mic close to the bell as is the norm in live situations.
Mic position will have a great effect on the sound you record - pointing the mic straight down the bell will always sound the brightest, but if you move very close to the mic, the proximity effect will increase the bass response of a lot of mics. Less so with the RE20, which was primarily designed as a radio dj's mic and has little proximity effect so the timbre of the voice doesn't change if you speak up close to the mic. It's also ~£360 new. The RE20's popularity in broadcast studios as a rugged mic that can handle anything probably explains why you often see them in TV clips of sax players
I've used M201's on other instruments and they've been great, but not had one for sax.
Mic-ing for live sound tends be a matter of feedback avoidance and getting as much signal down the mic with as little spill from the other instruments onstage. SM57's and 58's are good at these things, but they're not exactly high fidelity - I've never liked them, but it's usually all that's available. I've ceased caring what my sax sounds like if there's a PA involved. Getting paid, getting fed and getting a good night's sleep are my major ambitions when touring, but that's another subject...
The Beyer M69 and M88 are also good mics, the AKG C1000 is disappointing
If you're looking at secondhand M201's, some of the older ones may have Tuchel connectors rather than standard XLR's, but adaptors are available. This happened to one of my friends who bought a pair of M160 ribbon mics and had to get adaptor leads. Tuchels look like XLR's but are smaller, so check before buying. I think it's the M201N that sometimes had these connectors, probably for German broadcast use. The M201TG is ok.