Are you trying to achieve this sound "live in the room" or at recording?
A couple of hints for both…
You can tune ring in & out of a snare same as with toms, by finding one of the optimum resonance bands then dropping one corner to pitch it out of resonance. You then tune the bottom head to optimise the snare decay. Bottom snare head doesn't really affect the top head pitch in the same way as on a tom, but if it's sympathetic it will ring & the snares will buzz for longer than if it's in an 'anti-node' type tuning.
If you have to put tape on it, it's not tuned properly;)
A slight exaggeration, I carry a ½" piece of gaffa with a single layer of paper tissue in the centre in case I get a slight over-ring :p
Also note that where you hit the snare & at what angle will affect the ring. Smack dead centre with a high stick angle will be massively damped compared to a low angle & hitting just beyond the centre.
You can even remove apparent attack transient by using heavier sticks - even if that sounds counter intuitive. Light sticks & a shallow angle can be really 'whippy' whereas heavier sticks give a fuller body.
After that comes the mic technique.
The mic in the recording linked sounds to me like it's quite a way off the head & perhaps pointed slightly to the skin edge. I'm not hearing a bottom mic, though it might be in there & pulled way back in the mix. It's hard to tell precisely because of the additional ambience on the kit. Part of that 'distant mic' sound may be coming from the overheads or ambients rather than the actual snare mic itself.
You can do some trial & error - move the mic to different distances, from different directions & pointing at different areas of the head; from sneaking up just under the hi-hat, to way over the drummer. One interesting alternative which is very ringy & resonant (in a good way;) yet with highly detailed transients, is to sneak an omni in between the gap where all the heads line up, set just slightly higher than the intersection of the planes of all the top heads, snare & toms. If you get that right you can mic a whole kit with two close mics and two ambients, removing a whole lot of the need for gating or fighting with phase discrepancies.