I vote for the binomial names over common names, in any kind of seed or plant trading activity.
I'm holding a collection of donated seeds, and often something will be contrbuted with a one-word or two-word common name. In order to give the seeds away intelligently, I like to know things like "annual or perennial" and hardiness zone, and whether it's a flower or a shrub or tree.
However, for most common names, a short search will find dozens or hundreds of species, many hundred cultivars, and often as many as 5-6 different genra!
When I know who who contributed the seed, I ask gently if they have any more details for the name, or a description that might narrow it down some. However, one canny gardener warned me that many people would stop contributing if asked "for that much detail".
And one person made it very clear that she didn't care about names, to her the only improtant thing was that she liked how it looks.
It's true that not everyone is in love with details and fussy precision.
Personally, I wish there were a CSI or NCIS for plants that would do plant DNA tests and tell me exact species, cultivar, parent strains for recent hybrids, and which hillside that exact seed germinated on! But I doubt that would give any actual advantage to most normal, sane gardeners.
On the other hand, there is some advantage to knowing whether a traded seed is an annual flower or a 30 foot tree before you plant it!