I must gently disagree with @Michael's first comment. There would certainly be a significant shift in what we could say about convex optimization problems if we relax the definition as the OP suggests we might: we suddenly would not be able to claim that convex optimization problems, as a class, are tractable.
The most important reason we study convex models is that, unlike the larger class of mathematical models, they admit a variety of tractable numerical methods for solving them. And if I'm going to get beyond a theoretical paper to the actual solution of a given class of problems, I'm going to need describe them in a computer-friendly manner. For instance, I might write code to compute the values and derivatives of the objective and constraint functions. I might express the problem in conic form and use a barrier or scaling point method. Or I could rely on a cutting plane oracle or barrier function to describe the constraint set.
But if I were start to allow any sort of constraint combination you wish, then the likelihood that I will find such a computational description drops considerably. Consider the constraint $x^2 \geq 1$---it has no cutting plane oracle, and its barrier method is not convex; heck, the set it describes isn't even connected. But wait, if I also happen to know that $x\geq 0$, then I still have a convex problem, right? Well, sure, but of course the first thing I will do is transform my problem into a solvable form by replacing $x^2\geq 1$ with $x\geq 1$. I'm back to my comfortable Boyd-convex domain.
Sure, there are going to be some examples where a "non-convex" description of the problem still admits a tractable solution method. But hopefully we would agree that exceptional cases do not provide a useful foundation on which to build an entire practice.