Why running tap water over mussels not kill them, but soaking does? Why doesn't rinsing mussels with tap water kill them like soaking does? I know running water over mussels takes very little time, compared to soaking them, but the water is still entering the mussels.
For example, see this quote from Knorr UK: How to Clean Mussels:

Keep the cleaned mussels under the running water. It’s really important that the water stays moving over the mussels, but that they’re not soaking in it. If you leave them soaking in tap water, they’ll die and that’s not what you want.

This other quote from Serious Eats: How to Clean and Debeard Mussels implies that some mussels should be soaked--does soaking kill them or not?

[F]arm-raised mussels are held in tanks prior to packaging and shipping, which means that the purging step—soaking the mussels in clean water until they spit out impurities—has already been done for you
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:

Don't soak in water
Don't immerse them in water - fresh or salt. Freshwater will kill them; if left for too long in static salt water the mussels will use up the oxygen and suffocate. This practice used to be done to purge the mussel of any grit, these days all commercial mussel have been purged and purified in a UV deputation system.

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and The Cornish Mussel Shack (and many other places on the web) is that soaking (whether in fresh- or salt-water) is an outdated practice unless you're using wild-caught mussels (presumably including ones you've collected yourself).