From an in-character perspective, the term doesn't apply very well at all.
There are no common items, just items.
I suppose I simply disagree. Now, yes, you have a point that every setting and campaign may be different. But I think the goal here is that characters do indeed recognize some items as common compared to others.
"Sure, enchanters all over the kingdom know how to make magic swords. And yeah, everyone's heard of elves and their magic cloaks that let them fade into the woods. But you say you have a sword that shoots boulders? Wow! I've never heard of that before!"
How true such things are will vary from setting to setting. But clearly the goal is that there is in-character recognition of common items, and it certainly fits with the genre and past versions of the game. It's just that, in the past, how 'common' an item was tied into its price more than anything else. Now, it's tied to something else instead.
I think if WotC had this system in place from the start, most of these issues wouldn't exist. It all has come about from trying to force it into place on an existing system without having yet provided full support for the new system.
But again - all that aside, the term is certainly reasonable in character. Common items are ones that many adventurers often have, that average folk have heard of, etc. Uncommon ones might be largely unknown, or more limited in lore, or just the stuff of folk tales or legends. And rare items will be truly mythic in scope, much like artifacts.
(Which, again, I think WotC has undercut by tossing the system out when they had so few good examples. Gauntlets of Ogre Power really don't seem appropriate for that. Whereas the recent Dragon article on the Ring of Winter, Orbs of Dragonkind, etc - that shows a better look at the picture.)