I don't believe that either of these points are true. In your original example, there is one correct solution for any convex function. I will assume there is a single hard-coded function for the following, but it can be extended to work for an arbitrary function.
The output register having the correct solution is the target set.
The output register having any state is the basin of attraction.
Clearly any specific number (or rather singleton of that number) is a subset of all numbers, so the target is a subset of the basin. And further, because "all numbers" has more than one element, the target set is smaller than the basin.
This end state state is the solution to the convex function being stored in some physical registers. The initial state is those registers containing arbitrary data to be overwritten. It's not particularly interesting as optimization problems go (not a very large basin of attraction) but it fulfills the basic criteria.
The unique thing about your example is that it solves once and then it is done (relative to the examples in the post), so it ceases to be an optimizing system once it finishes computing the solution to your convex function.
With a slight modification, you could be repeating this algorithm in a loop so it constantly recalculates a new function. Now the initial state can be some value in the result and input registers, and the target region is the set of input equations and appropriate solution in the output registers. It widens the basin of attraction to both the input and output registers rather than just the output.
Pretty much, yes, according to definition given. Like I said, not a particularly interesting optimization but an optimization none the less.
To extend on this, the basin of optimization is not any smaller than an iterative process acting on a single register (and if you loop the program, then the time horizon is the same). In both cases your basin is anything in that register and the target state is one particular number in that register. As far as I can tell the definition doesn't have any way of saying that one is "more of an optimizer" than the other. If anything, the fixed output is more optimized because it arrives more quickly.