The rules don't actually define cover at all. They provide guidelines to the DM.
In fact the rules hardly define anything. But we can tease out a definition if you want.
The cover rules tell you generally what kind of thing cover is, and then how to represent it in the game mechanics. "Walls, trees, creatures" are examples to illustrate what kind of thing we're talking about. The catchall category of "... other obstacles" implies that all of these are obstacles. An obstacle is something that obstructs, blocks a path, gets in the way.
The other important words here are "in combat" and "making a creature more difficult to harm." This tells us what property of the obstacles enables them to act as cover (that they make a creature more difficult to harm) and the context where that property should be considered ("combat", so the harm we're concerned with is getting punched in the face or shot with arrows or something like that).
Putting this together, cover is an obstacle that makes a creature more difficult to harm in combat.
So if you put up a silk curtain between you and the ax-wielding Orc berserker, is that going to make you more difficult to harm? Probably not, so it's not cover in that situation.
On the other hand, if the thing trying to harm you is a swarm of stirges, a silk curtain or even a sturdy net might keep them out.
Another angle to consider: I could see a ruling treating "soft" cover as a lesser degree of cover, even if it completely obstructs the target. Mechanically, partial cover adds to AC, and AC includes not just things that prevent attacks from hitting (like Dexterity or having a shield) but also things that make attacks less likely to hit a vital spot (such as plate armor) or reduce the force of an attack (such as leather armor). So maybe you can shoot an arrow through that curtain, but it's going to get deflected to some extent, right? Remember that half cover is only a 10% decrease in the chance to hit. If you think there's a 90% chance that an arrow would punch through and hit the target, that's half cover.