What a good question.
Does a snare that cinches tighter cause more damage? No.
Generally, if the lock tightens up good, the snare remains in one place. If the snare moves around very much it is going to cause damage. The lock can pull fur. Also if it doesn’t tighten down sufficiently, the cable can “roll” into the fur and make a real mess.
Some of the damage from a snare, and some of the marking of the leather depends primarily on how much and how hard the animal has “struggled” in the snare. Lethal snares mark less than nonlethal snares. With nonlethal snares, animals that do not entangle often show less markings than those that do.
I’ll use the example of a beaver here. If you snare a beaver, and allow it to get back into the water where it feels safe, it will struggle less and have less of snare mark, than one you snare and hold on dry land. The beaver on land is struggling all night long to get back into the water.
Mind you, I don’t think any of this “fur damage” or “marking” would cause any serious long term injury for these animals if they were released, but it does reduce the current value of the pelt.
I really think this is again one of those circumstances where the lock actually plays a secondary role to the circumstances at the set. On an ending note, I do believe if you dispatch a live animal in a snare, you will get less marking if you remove the snare from the animal immediately after it has expired.