I find this to be more a philosophical question. To classify each type of interface is more likely a combination of terms. We would first have to think of all the different classifications that can be applied. There are some great examples listed above and choosing one would the first step. For instance, is it a website or a web application. What are the rules for classifying one vs the other.
Me personally I go by number of interactions. Which would lead me to believe interaction type(s) would be a secondary category. Of course this is my preference as I recently took a job to redesign a website and found myself neck deep in a web application. The difference to me being instead of dealing with the usual 10 - 20 interactions I was working with well over a hundred and variable rules for each.
So relating back to Andy's example of content style, does this include the back-end as well. Is it a CRM, CMS, Feed Based, Static Content etc.. Before running into this web app, this was my top priority for identifying what I was in for.
I may be making this more complicated than it needs to be, but what can I say, I like the question.
Sticking with just the UI as originally asked I am reminded of the recent surge in articles pertaining to classifications of UX Jobs. At first the descriptions were vague and more or less opinions on what they should be, but as these conversations grew more detailed and specific descriptions were provided.
I would imagine the same could be accomplished with UI. Me personally though, it's rare that I would concern myself with anything other than; responsive, dynamic or static to start. My next concern would be content; portal, feed, eCommerce etc.. and any combination there of. If I were to take it a step further, I may start to describe layout; fixed width, single column, two column, etc.. When I try to classify I use terms that are easily digestible by those who know very little about what I do.