Why running tap water over mussels not kill them, but soaking does?

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The Cornish Mussel Shack (and many other places on the web) is that soaking (whether in fresh- or salt-water) is an outdated practice unless you're using wild-caught mussels (presumably including ones you've collected yourself).

Mussels are a saltwater creature and immersion in freshwater for an extended time will kill them. For example, from Dieplicious (under 'CLEANING THE MUSSELS'):

It is a common misconception that you should soak the mussels in freshwater for an hour to purge them from any grit, but this will actually kill them, as they can’t handle the freshwater. If you buy farmed mussels (such as from Prince Edward Island, or PEI), then grit should not be a significant issue. However, if you purchase wild mussels, you should soak them in cold water (in the refrigerator is fine) with a generous amount of salt added for not much longer than 20 minutes, to allow the grit to purge out of the mussels.

[Added]:

Mussels are normally intertidal. When the tide goes out (i.e., they're not surrounded by water), they close up, sealing themselves off from the external environment. While rinsing them off, they are not interacting with the water (except for their shells). If they are open during this process, they are usually dead; hence the common direction to discard them.