As for the other end of the question, "no ties," I think that Peaches's point about stating things in the negative is spot on here. "Ties not required" is much better. Even if most people would figure that "no ties" means "ties not required," they wouldn't be 100% sure, and you can bet that if you wear a tie, there'd be at least one idiot telling you, "Hey, the invitation said no ties!" (The conversation that would follow would resemble this string!)
I guess this is a minority opinion, but I completely agree with you and peaches on the negativity issue. It reminds me of invites I get that say "adult reception only". The word "only" hits me wrong because it sounds exclusionary but, more important, it's redundant! "Adult reception to follow" says the same thing.
Anyway, I'd think there has to be a term among your region/age group/whatever that would work to describe this. In my neck of the woods, this would be business casual. And if I ever got something that said "no tie", I'd take that literally to mean don't wear a tie, not as "tie optional". And that becomes a strange request.
While I definitely agree with you - OTOH I could understand (not condone, but understand) a HC using the term "adult reception only" if they know-their-audience, so-to-speak. Some clueless folks really do need to be hit-over-the-head with stuff like this, in a manner of speaking. They might interpret "adult reception" to mean "children optional". The term only would get the message across more firmly. Of course there are no guarantees - we all know parents who would interpret this as meaning "all children except their children"...
And I would find it kind of confusing, at least at first. I admit that when I saw "adult reception only" I thought for a moment it meant that adults were only invited to the reception; children were invited to both the ceremony and the reception! I figured it out quickly, and I'm sure all guests would. I think it's an awkward wording; maybe they were trying to avoid the more syntactically natural "adults-only reception" because it sounds like it's rated X or something?
But even if there aren't ultimately going to be any mistakes, you don't really want any of your guests' first impression to be, "Huh? Oh, I get it."
Mikayla, I think you made an excellent point (bolded)! There may not, at least not yet, be an etiquette-recognized term equivalent to "black tie" (although we know even that has regional variations). But I bet you're right that in most cases there will be a term that you know that your invitees will understand. Around here, it would be "dressy casual." I don't like it, lots of us here don't like it, but there it is, and I think that if these hosts had used it, they'd have gotten what they were aiming for. Perhaps a few people in ties, but not with a jacket, let alone a suit, and maybe even some denim, but only as part of a very appropriate, dressed-up-enough-for-a-nice-restaurant outfit. But so what -- they would have gotten the look for their party that they want, and that's really as specific as they should be. Other places, other terms would be commonly understood.
If there are a few people that you think need more guidance to understand something like "casual" not meaning sloppy T shirts and cutoffs, you can somehow get into a what-are-you-wearing conversation sometime if they live nearby, or for those who don't, perhaps an email to the effect of "Just as a heads-up -- I know the term "casual" varies so much by community; around here it means a step up from jeans, but no ties necessary."