Is a binding ZLB a binding constraint? Usually, in an optimisation problem, a binding constraint is one at which the optimal solution holds at the constraint with equality, i.e. it's a boundary solution.
However, in many articles, for example this one, from what I could interpret, the authors talk about non-binding ZLB as if meaning that the ZLB could be not valid, and the interest rates be negative (like in Switzerland and Japan). When talking about a binding ZLB, they seem to mean that ZLB is valid.
To me, at least, this seems to be at odds with the usual definition of a binding/non-binding constraint. Or am I missing something?
I think this is due to how terminology surrounding ZLB changed/evolved over time.
In most models ZLB is actually mathematically zero lower bound in a way that is closer to a boundary solution as you put it (see examples in Romer Advanced Macroeconomics chapter 11 and more importantly sources cited therein for the models including ZLB). Nonetheless, the ZLB result hinges on many simplifications such as that there is no cost to holding cash money. Such assumptions might be fine for model building purposes but when it comes to practical policy making those storage costs can easily result in the effective lower bound being let's say $-0.5\%$ or some other value that would have to be empirically estimated.
However, in the mean time zero lower bound somehow acquired separate meaning as being the term for the idea that interest rates cannot fall beyond zero, even if originally the terminology coincided more closely to what mathematicians would consider biding constraint. Colloquially, at least from my own experience, when people discuss ZLB at conferences/workshops they often do it in a sense of referring to the result or even sometimes just as referring to idea of lower bound rather than trying to follow mathematical terminology.