I think you have it right, though you can do either with any bike. However, if carrying a heavy load, a "touring" specific frame would be ideal. And if you are "competitively" randonneuring, you want a fast bike.

Jan Heine detractors are quick to point out that most of the top finishers of the Paris-Brest-Paris brevet ride carbon fiber performance bikes. Most seem to miss his point, however, that the top finishers probably wouldn't be significantly slower on a traditional randonneuring frame, and in fact with the comfort and control afforded by wider tires, it's possible they could finish in a faster time.