The rated wind speed is usually defined as the minimum wind speed at which the rated power (i.e. maximum power) is produced.
The design wind speed is a slightly less clearly defined concept, but often refers to the wind speed for which the design of the turbine is optimised. The speed to choose for this might, for example, be the most commonly encountered wind speed, or (more likely) the wind speed that over the course of an average year provides the most energy, or a more complex mixture of these two convolved with turbine performance that results from an engineering optimisation process.
A tip speed ratio of 7 is in the range of good TSRs for modern wind turbines, so it's entirely possible that a given turbine design will have a TSR of 7 at its design wind speed - it's even possible that this is one of the design objectives - but I'm not aware of the design speed being defined that way. I don't commonly work in wind, so my knowledge is not exhaustive, i.e. this doesn't mean that it isn't defined like that by some people.