Since there have been no answers in five days, perhaps I can give a partial answer. I am by no means an expert, so I hope someone will point out my mistakes or if there is any bias (which I am sure there is). Also, my knowledge is limited to political science, but still I hope there is something that you can get from this.
In Britannica, structural functionalism, is defined as "a school of thought according to which each of the institutions, relationships, roles, and norms that together constitute a society serves a purpose, and each is indispensable for the continued existence of the others and of society as a whole." So for Britannica, the view seems to be about mutually interdependent structures in society. According to wikipedia, a common analogy likens these structures to "organs." The wikipedia article also lists a lot of criticism to this approach. To give an example of "real research" along these lines; many view the Varieties of Capitalism-framework (Soskice & Hall, 2001) as functionalistic (but you should make up your own mind about that, of course ;)).
A common critique of structural-functionalism is indeed that it explains the existence of institutional arrangements based on some notion of efficiency or optimality. This critique seems to be very common among historical-institutionalists in political science, who stress that institutions are the products of historical processes, and even in cases where they are the products of some group of actors to solve a particular problem, they can often "outlive" their creators and, with time, change to serve entirely new purposes.
An example of this can be found in Kathleen Thelen's critique of the structural-functionalist perspective. In her book How Institutions Evolve (2004), she describes (among other things) how the German system of vocational training evolved from being a system initially designed to weaken the position of the labor movement, to the diametrically opposite position of being a central pillar in the cooperation between the state, employer associations and labor. I recommend reading the introduction and following the references therein - she also references research using the functionalist perspective!
Pierson (Politics in Time, 2004) defines a "functional interpretation" as a view where "institutional arrangements are explained by their outcomes," or where "institutions take the form they do because powerful actors engaged in rational, strategic behavior are seeking to produce the outcomes observed." This he calls "actor-centered functionalism," which he distinguishes from "societal functionalism," in which institutions exists because they fulfill a certain function for society as a whole.
For Pierson, functional accounts are seen as inadequate for historical research, as they are unable to account for "lag" between the moves of actors and actual institutional outcomes (among other things). I recommend reading Piersons introduction and chapter 4 and 5 if you are interested in what he says about functionalism (I really recommend the entire book, actually :)).
As for my subjective view (for whatever its worth to you ...), I'm not really sure what functionalism really is. I have read many articles that analyze institutions or human behavior using game-theoretic models and so on, and in my (narrow) view, this literature represents the stereotype of functionalist accounts. Even so, there are works that utilize game-theory in historical research (see especially: Greif, Avner, 2006, Institutions and the Path to the Modern Economy) and that tries actively to overcome the sort of criticism that I have cited above, so there is certainly a blurry line of demarcation here.
I am sorry for the heavily biased reply to your post, and I am not entirely sure it was the answer you were looking for. Reading the critique of researchers in the "historical institutionalism" camp will certainly answer your question positively - functionalism is indeed about purposefully designed institutions (but, as Pierson points out, this does not mean that the perspective is without merit!).
Moreover, due to my own narrow perspective; what researchers subscribing to functionalist views themselves think, I do not know. Also, if there is any difference between sociology and political science I cannot tell. Hopefully someone will reply with a more informed view if they feel this reply is inadequate :)
