The President's Choice mussels in sauce are (properly) fully cooked and then frozen in packages for your convenience. I can't find heating instructions for their products but, regardless of which method you use, they should be prepared from frozen with the goal being to heat them - not cook them.
Regarding the live PEI mussels, you are overcooking them. If the meat is shriveled, they are overcooked. You can't just assume that 5 minutes is the ticket for properly cooked mussels.
While there are different methods of cooking mussels and other shellfish such as grilling or baking, for steaming, there are some tips to help ensure that the mussels are fully cooked but not overdone.
The reason for letting them sit for a minute or two in the residual heat is that some mussels will open very quickly, ~3 minutes. That typically is not considered enough time for them to be fully cooked. And if they take longer, an extra minute or two in the residual heat is not enough to ruin them.
Last but not least, remember that time can vary. Even with mussels that take longer, it's still a quick process. So it's important to keep your eye on things.
Where I come from (Portugal) it is quite common to see in most restaurants and establishments shellfish being soaked several hours to even days long before consumption, especially sand dwelling shellfish, which is not the particular case of mussels.
The reason for this is that it keeps them alive and fresh for longer, while also purging any remaining sand or debris naturally found inside them, which is very common in burrowing shellfish, quite unpleasant to find when eating and can virtually ruin the dish.
These are however always soaked in either salted tap water at worse, or at preferably clean natural sea water from where they came from at best. Never just unsalted tap water, because this would obviously kill them quickly, and remove any natural salt that acts as flavor enhancer. Chlorine based disinfectants commonly present in tap water will also slowly affect quality of any living creatures (including aquarium fish) and may eventually be fatal them in the long run.
From your provided examples, most against soaking either state that tap water will kill them quickly, or considerably affect quality (which are both true); or is not needed for farmed shellfish.
I can't speak much for farmed shellfish, they are not as common here and quality may vary with providers and techniques, but even farmed one can some times be quite sandy.
So I'd say that soaking with either salted water or ideally sea water is at at worst redundant or not needed, but can be quite beneficial in some situations.
For farmed shellfish, if you find them clean and edible it is probably not needed, for "free range" ones you probably have more to gain by soaking than not.
Best Answer
Rinsing them doesn't kill them because a small amount of fresh water isn't toxic to them and, even if it were, the mussels aren't going to absorb much from a quick rinse. I've not seen this instruction about keeping them under running water other than to clean them. It makes sense during the scrubbing process but not as much as a holding method since mussels are just fine out of water for a few days, let alone the time between cleaning and cooking.
The line in the Serious Eats quote doesn't mean that the mussels are soaked in fresh water. It's clean water - meaning water free of silt. But it's still saltwater and it's done professionally, knowing how to preserve the mussels - which someone in a home may not.
A company that sells mussels offers the following:
You'll note the emphasis here is static salt water. Moving, oxygenated salt water is fine but people don't generally have the ability to do this in their home (I suppose unless they have a saltwater fish tank).
The end result is - lots of people have different methodologies for cleaning mussels. What it sounds like from both Serious Eats and The Cornish Mussel Shack (and many other places on the web) is that soaking (whether in fresh- or salt-water) is an outdated practice unless you're using wild-caught mussels (presumably including ones you've collected yourself).