History tends to teach us that attempting to design something without having a clear idea of what you ultimately want to achieve rarely works out well. Equally the definition of the problem to be solved should address the underlying problem, not just be an outline of a proposed solution.
There is, of course, a perfectly valid place for open-ended or speculative research and development, but this isn't quite the same thing as actually designing something.
Often, if the requirements of a design seem very flexible it's because they haven't been defined well enough. The requirements should define what you want to achieve not a wishlist of 'features' which may or may not assist in solving the core problem.
Of course context matters a lot. For example if you are a member of a design team it may well be that another designer will give you a very specific set of specifications. Say you are designing an alternator for a specific engine the lead designer may well tell you specifics like maximum current capacity, duty cycle, average current capacity, voltage tolerance, maximum mass, environmental durability metrics and life expectancy.
On the other hand if you are developing an overall design concept for a car you might be negotiating rather more diffuse goals with strategic marketing people who might very well have rather more abstract objectives in mind and it is when it comes to 'designing' a marketable product rather than a strictly engineering a solution to a well defined problem.
This ultimately comes down to the fact that the goal of 'what can I design that people will buy at a profitable price for me' is not a strictly engineering problem. In the same way that 'how can I improve the reliability of this pump without increasing its cost too much', very much is.
Similarly anyone who has worked in any kind of freelance capacity will be well aware that what a client says they want is often a statement of how they imagine the problem should be solved rather than a definition of what they actually want the design to achieve.
I think that there is a good argument that one of the defining skills of engineering is the ability properly identify what the real problem/need is without presupposing a particular solution and similarly to be able to differentiate between problematic symptoms and their underlying cause.