In the wild west of early America, 2 accusations were seen as existential threats that warrants shooting someone, even if just implied and not backed down from when called out:

1) being called a liar

&

2) being called a coward

In those days, there wasn't much law. Millions of dollars of business (in today's dollars) and multi month contracts of employment were settled on handshakes. A LOT of weight was placed on knowing "who you could ride the river with" and wouldn't bail when food got tight of the trail / would stand strong when indians or outlaws attacked.

Those 2 accusations mattered because they indicate you are not a man who can bear weight of being trusted, thus you lose all access of employment and business whether as a leader or a worker.

That reputation getting around about you would literally ostracise you from the western economy and destroy your livelihood, thus people took it VERY seriously. They were seen as a killing offense

And it was warranted.