It’s very difficult to prove a negative, but I am reasonably confident that no rule in 3.5 or Pathfinder explicitly states that all armors of a given weight class slow you down by the same amount. In 3.5, at least, there were certainly a few super-heavy armors that reduced the speed by more than your typical heavy armor (Races of Stone had a couple of these).

So yes, every Medium or Heavy armor must somehow explicitly state its own speed reduction, even if the overwhelming majority of them have the same speed reduction. On the other hand, if a given case failed to do so, I’d just chalk that up to the designers not realizing there was no such general rule, since it usually is so consistent and armors which slow people more or less than others in their category are quite rare. It’s also worth keeping in mind that it is consistently weight category that we should consider, not sheer weight.

As far as the rules are concerned, you've already quoted everything that is relevant here:

If the Armor table shows "Disadvantage" in the Stealth column, the wearer has disadvantage on Dexterity (Stealth) checks.

There is no caveat there about "while moving" or "except when standing still", so yes, the penalty still applies when the wearer is standing still.

You are (as always) free to rule otherwise in your games, or if you're a player, ask the GM to rule otherwise. Of particular note here is the description of Dexterity checks:

A Dexterity check can model any attempt to move nimbly, quickly, or quietly, or to keep from falling on tricky footing.

So if you're standing still, the check might not be a Dexterity check. The description of the Stealth skill doesn't contradict this, either:

Stealth. Make a Dexterity (Stealth) check when you attempt to conceal yourself from enemies, slink past guards, slip away without being noticed, or sneak up on someone without being seen or heard.

Nothing here suggests that standing still should be modeled as a Stealth check, so it could be worth asking your GM (or ruling, if you are a GM), for some other kind of check to model this. The section (I won't quote it all here) on using skills with different ability checks (PHB pg. 175) may be useful for supporting this.