I think you are asking a question that really has no suitable answer. You would do as well asking Alan Parsons how he captured that sound as you would asking Nick Mason how he tuned his snare.
As mentioned, odds are he used a coated Ambassador for his batter head and whatever may have been on his snare side at the time. The head(s) may have been old or may have been new. I am sure there was some muffling involved, but possibly not as much as you think. I only have access to "Money" currently, but that snare does not sound all that detuned to me. Your initial track was fairly close, but I would agree it is a pretty squashed which is typical of the time period, but maybe a bit much in your case, especially when you include other instruments over the top.
To that point, we often hear drum sounds in the context of an entire band playing and we somehow imagine the sound that we hear is the actual sound of the drum in the recording. In reality, if you could solo up the snare drum or bass drum, you would generally find that it was a bit or even a lot more complex than what you originally thought. Of course, in the 70's we did tend to completely kill the sound of an actual drum but perhaps I should not go there as I do have strong feeling on that subject.
Most drummers will spend a lot of time trying to tune their drums to sound like their favorite album. The reason I say this quest (as it were) may have no suitable answer is simply because acoustic drums in a room, generally do not sound like recorded drums in the booth. If you walked into any recording studio and listened to the engineer's monitors and then walked out onto the tracking floor, that sound will be very different. You would go from a very detailed controlled sound to a blitzkrieg of high end and general wash. It was true in the 70's and it is true today. If you listened to the playback of DSotM and walked onto the tracking floor to stand next to Nick Mason, it would be a very different sound. Unless you have that snare drum, with those head and snare combinations, tuned the same, placed in the same spot of the same room, with the same mics, mic placement, desk, outboard gear, tape machine and you happened to be Nic Mason and you had Alan Parsons engineering, then it still wouldn't be exactly the same...but it might be close.
If I were trying to more or less replicate that sound, I might consider a head with a bit of muffling to start with. Perhaps a Remo coated CS or an Evans Genera Dry HD. My guess is that the batter head was tensioned in it's mid range. I have no idea about the snare side head, however, I generally favor a pretty tight snare head. Try a 42' strand snare, a 20 strand and a 16 strand and see what YOU like. From a quick listen to "Money" there was a decent amount of actual snare wire in the snare sound, but that may have nothing to do with the number of snares and everything to do with the placement of the bottom mic in the mix and the way it was eq'd.
As Oddball mentioned, you also need to consider your playing. The drum will sound quite different when played at different velocities, differing stick angles and technique (playing into or out of the drum for instance). It will also sound very different depending on the type of stick you use. A heavier stick with a broader wood tip will produce a much beefier sound than say a 7A with a nylon tip. The only way to create a consistent sound is to play consistently. Every backbeat needs to be played at the same velocity and angle unless, of course, there are some dynamics that come into play. Most pop and rock music relies on a drummer that can play a consistent back beat and that can be far more difficult than most imagine. Every stroke matters, and in my opinion, that will supersede any bit of head, snare, shell, tuning, mic selection, eq, processing or record media any day of the week.
As to live playing, keep in mind that audio in a live setting has the issue of exciting tens of thousands of cubic feet of ambience as opposed to the dozens of cubic feet in one's living room, car, or or the zero amount of cubic feet in one's headphones. In general, detuned drums just do not cut it live, especially back in the 70's when PA systems were a complete hodgepodge of bad ideas. Heck, at that point, audio companies were still stacking PA's as the idea of inverting a chain hoist to climb it's own chain had not been discovered so there was no reasonable option to fly those systems. Therefore, the hodgepodge of speakers were throwing their comb filtered and mis aligned output directly at the back wall of a venue creating an audio morass. Tuning drums up helped in letting them cut through those situations as Stewart Copeland had so correctly noted in the early days of the Police.
Hopefully this has been at least somewhat helpful. You can chase that sound forever, or you can develop a sound that you like, that fits the style of music you like and then enjoy your playing. It will never sound exactly like Dark Side of the Moon, but it can probably sound plenty close enough. Good luck.
