::This post really veers off from anything that might possibly concern a layman.  Rigorous formal definitions are given in the appropriate section, for someone to whom such issues might be a worry.  I agree that talking about pointwise properties of the delta function is simply "code" to mean where it is supported, but I think you'll find that (outside of very formalistic writings) it is a code that is generally used to talk about the support of a distribution, if not to rigorously define it.